UC-NRLF 


*B   531    Tfls 


A  REVIEW  OF  ALGEBRA 


BY 


ROMEYN  HENRY  RIVENBURG,  A.M. 

HEAD   OF  THE   DEPARTMENT   OF   MATHEMATICS 
THE   PEDDIE   INSTITUTE,   HIGHTSTOWN,   N.J. 


AMERICAN  BOOK  COMPANY 

NEW  YORK  CINCINNATI  CHICAGO 


copykight,  1914, 
By  EOMEYN  H.   RIVENBUEG. 

Copyright,  1914,  in  Gbkat  Britain. 

a  review  of  alget^ra. 
E.  P.  I 


PREFACE 

In  most  high  schools  the  course  in  Elementary  Algebra  is 
finished  by  the  end  of  the  second  year.  By  the  senior  year, 
most  students  have  forgotten  many  of  the  principles,  and  a 
thorough  review  is  necessary  in  order  to  prepare  college  candi- 
dates for  the  entrance  examinations  and  for  effective  work  in 
the  freshman  year  in  college.  Eecognizing  this  need,  many 
schools  are  devoting  at  least  two  periods  a  week  for  part  of  the 
senior  year  to  a  review  of  algebra. 

For  such  a  review  the  regular  textbook  is  inadequate.  From 
an  embarrassment  of  riches  the  teacher  finds  it  laborious  to 
select  the  proper  examples,  while  the  student  wastes  time  in 
searching  for  scattered  assignments.  The  object  of  this  book 
is  to  conserve  the  time  and  effort  of  both  teacher  and  student, 
by  providing  a  thorough  and  effective  review  that  can  readily 
be  completed,  if  need  be,  in  two  periods  a  week  for  a  half  year. 

Each  student  is  expected  to  use  his  regular  textbook  in 
algebra  for  reference,  as  he  would  use  a  dictionary,  —  to  recall 
a  definition,  a  rule,  or  a  process  that  he  has  forgotten.  He 
should  be  encouraged  to  think  his  way  out  wherever  possible, 
however,  and  to  refer  to  the  textbook  only  when  forced  to  do 
so  as  a  last  resort. 

3 


4  PREFACE 

The  definitions  given  in  the  General  Outline  should  be 
reviewed  as  occasion  arises  for  their  use.  The  whole  Outline 
can  be  profitably  employed  for  rapid  class  reviews,  by  covering 
the  part  of  the  Outline  that  indicates  the  answer,  the  method, 
the  example,  or  the  formula,  as  the  case  may  be. 

The  whole  scheme  of  the  book  is  ordinarily  to  have  a  page 
of  problems  represent  a  day's  work.  This,  of  course,  does  not 
apply  to  the  Outlines  or  the  few  pages  of  theory,  which  can  be 
covered  more  rapidly.  By  this  plan,  making  only  a  part  of  the 
omissions  indicated  in  the  next  paragraph,  the  essentials  of  the 
algebra  can  be  readily  covered,  if  need  be,  in  from  thirty  to 
thirty-two  lessons,  thus  leaving  time  for  tests,  even  if  only 
eighteen  weeks,  of  two  periods  each,  are  allotted  to  the  course. 

If  a  brief  course  is  desired,  the  Miscellaneous  Examples 
(pp.  31  to  35,  50  to  52),  many  of  the  problems  at  the  end  of 
the  book,  and  the  College  Entrance  Examinations  may  be 
omitted  without  marring  the  continuity  or  the  comprehensive- 
ness of  the  review. 

ROMEYN  H.  RIVENBURG. 


CONTENTS 


PAGES 

Outline  of  Elementary  and  Intermediate  Algebra  .         .  7-13 

Order  of  Operations,  Evaluation,  Parentheses         .         .  14 

Special  Rules  of  Multiplication  and  Division           .         .  15 

Cases  in  Factoring  .........  16,  17 

Factoring  ...........  18 

Highest  Common  Factor  and  Lowest  Common  Multiple    .  19 

Fractions 20 

Complex  Fractions  and  Fractional  Equations    .         .         .  21,  22 

Simultaneous  Equations  and  Involution      ....  23,  24 

Square  Root 25 

Theory  of  Exponents .  26-28 

Radicals •    .         .  29,  30 

Miscellaneous  Examples,  Algebra  to  Quadratics      .         .  31-35 

Quadratic  Equations 36,  37 

The  Theory  of  Quadratic  Equations  .....  38-41 

Outline  of  Simultaneous  Quadratics  .....  42,  43 

Simultaneous  Quadratics 44 

Ratio  and  Proportion 45,  46 

Arithmetical  Progression 47 

Geometrical  Progression 48 

The  Binomial  Theorem 49 

Miscellaneous  Examples,  Quadratics  and  Beyond     .         .  50-52 
Problems  —  Linear    Equations,    Simultaneous    Equations, 

Quadratic  Equations,  Simultaneous  Quadratics           .  53-57 

College  Entrance  Examinations 58-80 


OUTLINE  OF  ELEMENTARY 
AND   INTERMEDIATE   ALGEBRA 


Important 
Definitions 


Special 
Rules  for 
Multiplication 
and  Division 


Factors  ;  coefficient ;  exponent ;  power  j  base ; 
term  ;  algebraic  sum  ;  similar  terms  ;  degree ;  ho- 
mogeneous expression  ;  linear  equation  ;  root  of  an 
equation ;  root  of  an  expression ;  identity ;  con- 
ditional equation;  prime  quantity;  highest  com- 
mon factor  (H.  C.  F.)  ;  lowest  common  multiple 
(L.  C.  M.)  ;  involution  ;  evolution  ;  imaginary 
number ;  real  number  ;  rational ;  similar  radicals  ; 
binomial  surd  ;  pure  quadratic  equation ;  affected 
quadratic  equation ;  equation  in  the  quadratic 
form ;  simultaneous  linear  equations ;  simulta- 
neous quadratic  equations;  discriminant;  sym- 
metrical expression;  ratio;  proportion;  fourth 
proportional ;  third  proportional ;  mean  propor- 
tional ;  arithmetic  progression ;  geometric  pro- 
gression ;  S  oo, 

1.  Square  of  the  sum  of  two  quantities. 

(x  +  yy. 

2.  Square  of  the  diiference  of  two  quantities. 

(x  -  yy. 

3.  Product  of  the  sum  and  diiference  of  two  quan- 

tities. (sJ^t){s-t). 

4.  Product  of  two  binomials  having   a  common 

term.  (x^r){x  +  m). 

7 


OUTLINE 


5.  Product  of  two  binomials  whose  correspond- 

ing terms  are  similar. 

6.  Square  of  a  polynomial. 


Special 
Rules  for 
Multiplication 
and  Division 
(continued) 


Cases  in 
Factoring 


7.  Sum  of  two  cubes. 


■  ^y  +  y^' 


0^  4-  y^ 
x  +  y 

8.  Difference  of  two  cubes. 

— 11^  =  x'^-\-xy  -^  ?/2. 

X  -y 

9.  Sum  or  difference  of  two  like  powers. 

x'  -\-y'^    x^  —  y^    x^  —  y"^    x^~y^ 
x-\-y^    x~y'    x  —  y'   x-\-y 

1.  Common  monomial  factor. 

mx  4-  my  —  mz  =  m(x-i-  y  —  z), 

2.  Trinomial  that  is  a  perfect  square. 

x'^  ±  2  xy  -j-y"^  =  {x  ±  yf. 

3.  The  difference  of  two  squares. 

(a)  Two  terms,     x^  —  y'^  =  {x-]-  y){x  —  y). 
(h)  Four  terms. 

x^  -^  2  xy  -\- y"^  —  w?  =  {x  -\- y  -{-  m)(x  -\-y  —  m). 

(c)  Six  terms,     a?  -\-2  xy  -{-y"^  —  p^  —  2pq  —  q^ 

=  [(^  +  2/)  +  (P+g)][(^  +  2/)-(i>  +  ^7)]- 

(d)  Incomplete  square,     oj^  +  a;^?/^  -f  y^ 

= 0?^  4-  2  icy  4-  2/^ — ^V = (^^ + 2/^ + ^y){^^  -\-y^— ^y)' 

4.  Trinomial  of  the  form  x^  -{-hx-\-  c. 

aj2_5^_^6^(^_  2)(a;  _  3). 

5.  Trinomial  of  the  form  ax^  -\-hx-{-G. 

20aj2  +  7a;-6=(4a?  +  3)(5aj-2). 


OUTLINE 


9 


Cases  in 
Factoring 

{continued) 


H.C.F. 

and 
L.C.M. 


Fractions 


Simultaneous 
Equations 


Involution 


'  6.    Sum  or  difference  of 

'  two  cubes.     See  "  Special  Rules,"  7  and  8. 
two  like  powers.     See  ^-  Special  Eules/'  9. 

7.  Common  polynomial  factor.     Grouping. 

fp  -^fq  —  2  mp  —2  7nq 
=  tKP  +  g)  -  2  m(p  +  g)  =  (p  +  g)(^'  -  2  m). 

8.  Factor  Theorem. 

0^  +  17  a;  -  18  =  (a;  -  l){x'  +  oj  +  18). 

a2  +  2(i-3  =  (a  +  3)(a  -  1). 
a2+7a+12=(a  +  3)(a+  4). 
a^ -f  27  a  =  a(a  +  3)(a2  -  3  a  +  9). 
H.  C.  F.  =  a+3. 

L.  C.  M.  =  (a  +  3)(a  -  l)(a  +  ^)a{a^  _  3  a  +  9). 
'  Reduction  to  lowest  terms. 
Reduction  of  a  mixed   number  to   an   improper 

fraction. 
Reduction  of   an  improper   fraction  to  a  mixed 

number. 
Addition  and  subtraction  of  fractions. 
Multiplication  and  division  of  fractions. 
Law  of  signs  in  division,  changing  signs  of  fac- 
tors, etc. 
,  Complex  fractions. 

f  addition  or  subtraction. 
Solved  by  I  substitution. 
[  comparison. 
Graphical  representation. 

'  Law  of  signs. 
Binomial  theorem  laws. 

f  monomials  and  fractions. 
Expansion  of  {  binomials, 
trinomials. 


10 


OUTLINE 


Evolution 


'  Law  of  signs. 

Evolution  of  monomials  and  fractions. 
Square  root  of  algebraic  expressions. 
Square  root  of  arithmetical  numbers. 

.       ,    r  Cube  root  of  algebraic  expressions. 
I  Cube  root  of  arithmetical  numbers. 


Theory  of 
Exponents 


Proofs:  a^-a"  =  a"'+";   —  =  a"'-^  ;   (a*">=a'""; 
a" 


/^mn^^m.     (~j==p     {ahcY  =  a^h^C\ 


{  fractional  exponent. 
Meaning  of  \  zero  exponent. 

[  negative  exponent. 


Four  rules 


To  multiply  quantities  having  the 
same  base,  add  exponents. 

To  divide  quantities  having  the 
same  base,  subtract  exponents. 

To  raise  to  a  power,  multiply  ex- 
ponents. 

To  extract  a  root,  divide  the  expo- 
nent of  the  power  by  the  index 
of  the  root. 


Radicals 


Tr  an  sf  ormation 
of  radicals 


Radical  in  its  simplest  form. 

Fraction  under  the  radical  sign. 
Reduction  to  an  entire  surd. 
{  Changing  to  surds  of  different 
order. 
Reduction  to  simplest  form. 
.  Addition  and  subtraction  of  radicals. 


OUTLINE 


11 


Radicals 

(cojitiiiued) 


Multiplication  and  di- 
vision of  radicals 


■\/a  •  ^b  =  ^s/ab. 


f  Monomial  denominator. 
Rationalization  {  Binomial  denominator. 
Trinomial  denominator. 

Square  root  of  a  binomial  surd. 
Radical  equations.     Always  check  results  to  avoid 
extraneous  roots. 


Quadratic 
Equations 


fPure.     x^=^a. 

I  Affected,     ax^  -\- hx -]- c  =  0. 


Methods  of  solving 

Equations  in  the  quadratic  form 


'  Completing  the  square. 
Formula.     Developed  from 

ax'^  4-  6a;  +  c  =  0. 
Factoring. 


Properties  of  quadratics  - 


^2  =  - 


h 

,  ^/¥~- 

-4 

ac 

2a 

'          2 

a 

h 

V62- 

-4 

ac 

2a 


2  a 


Then  . 


b 
a 


Discriminant,  6^—4  ac, 
and  its  discussion. 

Nature  or  character  of 
the  roots. 


12 


OUTLINE 


Simultaneous 
Quadratics 


Case  I. 


Case  II. 


Case  III. 


Case  IY. 


Case  Y, 

Special 

Devices 


J  One  equation  linear. 
\  The  other  quadratic. 

(3x-y  =  12, 
\  x'-y'^^  16. 

f  Both  equations  homogeneous  and  of 
[     the  second  degree. 

(x^-xy  +  y^  =  21, 
\      y"^  —2  xy  =~  15. 

fAny  two  of  the  quantities  x-\-y, 
\  x^  +  y%  xy,  a?^^-2/^  a?—y^,  x—y, 
[      x^±xy  -\-  y'^,  etc.,  given. 

ra;2  +  2/'  =  41, 
I    0^  +  2/^9. 

'  Both  equations  symmetrical  or  sym- 
metrical except  for  sign.  Usually 
one  equation  of  high  degree,  the 
other  of  the  first  degree. 

I  x^  +  y^  =  242, 

\    x-\-y  =  2. 

'  I.    Solve  for  a  compound  unknown, 

like  xy,  X  -\-  y,  — ,  etc.,  first. 
xy 

xY  ■\-xy  =  ^, 
x-{-2y  =  —  5. 
XL   Divide   the  equations,  member 
by  member. 

0^-2/^=20, 
•T^  —  y"^  =  5. 

III.   Eliminate  the  quadratic  terms. 
4:X  -\-  3  y  =  2  xy, 
7  X  —  5  y  =  5  xy. 


OUTLINE 


13 


Ratio  and 
Proportion 


Progressions 


Binomial 
Theorem 


( mean, 
Proportionals  {  third, 
[  fourth. 

1.  Product  of  means  equals  product 
of  extremes. 

2.  If  the  product  of  two  numbers 
equals  the  product  of  two  other 
numbers,  either  pair,  etc. 

3.  Alternation. 

4.  Inversion. 

5.  Composition. 

6.  Division. 

7.  Composition  and  division. 

8.  In  a  series  of  equal  ratios,  the  sum 
of  the  antecedents  is  to  the  sum 
of  the  consequents  as  any  ante- 
cedent, etc. 

Special  method  of  proving  four  quantities  in  pro- 
portion.    Let  -  =  x,  «  =  bXf  etc. 


Theorems  < 


Let  -  =  x,  a- 
b 


'  Development  of  formulas. 


l^a-\-{n—V)d, 
/S'  =  ^[2a+(n-l)d]. 


ar"  —  a 


S  = 


S  = 


>^oo  = 


r- 

-1 

rl- 

-  a 

r  — 

1 

a 

1-r 


T        ^.         r  f  Arithmetical. 

Insertion  of  means  <  ^  ,  .     -. 

1^  Geometrical. 

Review  of  binomial  theorem  laws.    See  Involution. 
Expansion  of  (a  +  by. 

^.    T  ^        ,     f  key  number  method. 

Finding  any  term  by  <    ,,  ^      ,        . , ,,  ,  , ,     ^ 

&      J'  J  y  ^xh  Qj^.  (^.  _^  j^yh  ^3       method. 


A  REVIEW  OF  ALGEBRA 


ORDER  OF  OPERATIONS,  EVALUATION,  PARENTHESES 

Order  of  operations : 

First  of  all,  raising  to  a  power  and  extracting  a  root. 
Next,  multiplication  and  division. 
Last  of  all,  addition  and  subtraction. 

Find  the  value  of : 
1.  5.22- V25--5+2'. 8-4-2. 

2  28 


3.  9. 2-6  +  3-2. 42-- ■V/8-4  + 


3.22 


Evaluate : 

a^  —  a^  -{-¥  .  c-^/a  -f  a^hc  . 


4. 


5.    ^^^v'^  +  ^m,  if  ^  =  8,  m  =  27, 


,  if  a  =  1,  6  =  2,  c  =  3. 


^       2 V3  +  2  d  +  g  {^c-d)x      .^  ^ 

3Va  -\-h  —  ex—  G     1  ad~  Vabc 


a  =  5, 
6  =  3, 
c=-l, 
d  =  -2, 
x  =  0. 


7.  a-  \5b-  [a-(3c-36)+2cT-3(a-26-c)]J, 

if  a  =  —  3,  6  =  4,  c  =  —  5.  ( Yale.) 

Simplify : 

8.  m  —  [2m—  f3r—  (4r  —  2m)j]. 

9.  2a-  [5d+{3c-(a  +  [2(l-3a  +  4c])J]. 

10.    3  c2  +  c(2  a  -  [6  c  -  3  a  +  c  -  4  a]). 

14 


RULES   OF   MULTIPLICATION   AND   DIVISION  15 

SPECIAL  RULES   OF  MULTIPLICATION  AND  DIVISION 

Give  results  by  inspection : 

1.  {g  +  hW-  9.  ^=1^. 

c  —  d 
3   y  in     e^±A. 


3.  (2v-\-:^w){2v-^w). 

4.  {x-\-:dts){x-l  ts). 


10. 

e  +  d 

11.  tjzt, 
x-y 

12.       y . 

6.    fa-—  +  c-(^- 

^         ^  ^^  13.    (a -.03) (a -.0007). 

8. 


y^ -  27  P"^  15.    ^^-^'. 


y  —  Slc""  t  —  v^ 

17.  [(a  +  6)  +  (c  +  c^)][(a+6)-(c  +  d)]. 

18.  (p  —  g  +  r  —  s)(p  —  q  —  r  -\-  s). 

19.  (3  m  — n-Z  + 2  r)(3m  4-^-^  —  2  r). 

20.  (a;  +  5)(a;-2)(a;-5)(aj  +  2). 

21.  (a2  +  &2_c_2(^  +  3e)2. 

22.  f.  +  r-^  +  ^  +  .^Y.  23.    -'^^^ 


5         6  J  x  +  2 

References:    The  chapter  on  Special  llules  of  Multiplication 
and  Division  in  any  algebra. 
Special  Rules  of  Multiplication  and  Division  in 
the  Outline  in  the  front  of  the  book. 


16  CASES  IN  FACTORING 


CASES  IN  FACTORING 


The  number  of  terms  in  an  expression  usually  gives  the 
clue  to  the  possible  cases  under  which  it  may  come.  By  apply- 
ing the  test  for  each  and  eliminating  the  possible  cases  one  by 
one,  the  right  case  is  readily  found.  Hence,  the  number  of 
terms  in  the  expression  and  a  ready  and  accurate  knowledge 
of  the  Cases  in  Factoring  are  the  real  keys  to  success  in  this 
vitally  important  part  of  algebra. 

Case  I.    A  common  monomial  factor.     Applies  to  any  num- 
ber of  terms. 

5  ex  —  5  ct  -{-  5  CO  —  15  c^m  -h  25  G'm? 
=  5  e{x  —  t-^v  —  ?>em-\-5  e^m}). 

Case  II.    A    trinomial    that    is    a   perfect    square.      Three 
terms. 

di?  ±2  xm  -\-m^  =(x  ±  my. 

Case  III.    The  difference  of  two  squares. 

A.  Two  terms.     x^—if=ix-\-y){x  —  y), 

B.  Four  terms. 

x^  -\- 2  xy  -\-  y'^  —  w}  =  {x^  -^2  xy  -f  i/")  —  m^ 
=  (x  -\-y  -^m){x  -\-  y  —  m) 

C.  Six  terms,     x^  —  2xy-\-y'^  —  m^  —  2  mn  —  n^ 

=.(x^  —  2  xy  +  2/^)  —  {p^  -h  2  mn  +  n^) 

=  (cc  —  yy-  —  (m  -h  ny 

=  {_{x-y)-\-{m  +  n)']l(x-y)  -  (m  +  ^0]- 

D.  An  incomplete  square.     Three  terms,  and 

4th  powers  or  multiples  of  4. 


CASES   IN  FACTORING  17 

c4  _|_  c^fji}  +  (^4  ^  ^4  _|.  2  cVP  +  d'-  cM? 

=  (c2  +  (^2  _^  cd)(c2  +  cZ2  -  cd). 

Case    TV.   A  trinomial  of  the  form  x^  +  5x  +  c.   Three  terms. 

aj2_^^_30^  (a?  +  6)(a;  -  5). 

Case     V.    A   trinomial    of   the    form    ax'^  -{-bx-\-c.     Three 
terms. 

20x''  +  7  x-6  =  {4.x  +  3)(5  x  ~-  2). 

Case    VI.   A.  The  sum  or  difference  of  two  cubes.     Two 
terms. 

a;3  +  2/3  =  (a;  -I-  y){x^  —  xy  +  y^) ; 
a^  —  y^  z=  (x—y)(x'^  -\-^y  -\"  y^)' 

B.   The  sum  or  difference  of  two  like  powers. 
Two  terms. 

x^-y'^=(x-  y)(x^  -I-  x^y  +  xy"^  +  y^) ; 

x^  4-  2/^  =  (ic  +  y){x'^  —  o(^y  4-  x^y^  —  a?;^^  +  y"^). 

Case  VII.   A  common  polynomial    factor.     Any  composite 
number  of  terms. 

t^jj  +  fq  —  Pr  —  g'^p  —  g\  +  g'^r 
=  t\p-\-  q-r)-  g\p  -\-q-r) 

^(p-^q-r)(t'-g') 

=  (p  +  q-r)(t-i-g)(t~g). 

Case  VIII.  The  Factor  Theorem.     Any  number  of  terms. 
a^-\-17x-lS  =  (x-  l)(aj2  -\-x-\- 18). 

REV.    ALG.  — 2 


18  FACTORING 

FACTORING 

Review  the   Cases  in  Factoring  (see  Outline  on  preceding 
pages)  and  write  out  the  prime  factors  of  the  following : 

1.  Sa^^^am^l  11.  a;^"^  +  13  oj^"^  + 12. 

2.  x^-\-y\  12.  4  a262  _  (a2  +  &2  _  ^2)2. 

3.  4a;2_|_iia;-3.  13.  (x"  -  x- ^){f-x-20), 

4.  w?-\-n^  —  (l-\-2mn).  14.  a^  —  Sa  — a^  +  8. 

5.  -x^  +  2x-l-^x\  15.  jp3  +  7p2 -^  I4p  +  8. 

6.  x^^-y^\     (Five  factors.)  16.  1%  a% -^  ^  a¥ -\- bO  h\ 

7.  (aj  +  l)2-5i»-29.  17.  a?-lx^Q. 

8.  a;^  +  aj22/2  +  2/4.  18.  24.c'd''-4:lcd-lb. 

9.  a;^-llaj2+l.  19.      (^2  _  ^)2^)2  _  (^2  _  ^^>^2, 

10.    X2-  +  2-I-  — .  20.    aW-  —  -x^-\--' 

^2m  yZ  ^         yZ 

21.  gt-gk-{-gP-{-xt  —  xJc-\-xP. 

22.  (?7i  -  7i)  (2  a2  -  2  a&)  +  (n  -  m)  (2  a5  -  2  62) . 

23.  a2  -  a;2  -  2/2  4-  62  ^  2  a6  +  2  i»2/. 

24.  (2c2  +  3d2)a  +  (2a2  +  3c2)d 

25.  ^  ^^  -  ^)  a-^62  ^  (^  -  1)  (n  -  2)  ^,.3^3. 

1-2  ^         1.2.3 

26.  (x-  xy  -h  (a;2  -  1)3  +  (1  -  xy.  (M,  L  T.) 

27.  (27  yy  -  2  (27  y')  (8  6^)  4.  (8  b^.  {Princeton,) 

28.  (a3  +  8  63)(a4-6)_6a6(a2-2a6  +  4  62).      {M.LT.) 

Solve  by  factoring : 

29.  Q^  =  x.  30.    z^  —  4:Z-4.5  =  0.  31.   aj^  -  0^2  =  4  a;  -  4, 

Reference :  The  chapter  on  Factoring  in  any  algebra. 


H.  C.  F.   AND   L.  CM. 


19 


HIGHEST  COMMON  FACTOR  AND  LOWEST  COMMON 
MULTIPLE 

Define  H.C.F.  and  L. CM. 

Find  by  factoring  the  H.C.F.  and  L.C.M.: 


1. 

3  x^  —  3  a:, 

5. 

x'-2x^^x'', 

12^2(^--1), 

2x^-Ax^-4:X-j-6. 

18  0^(0^  - 1). 

(Yale.) 

2. 

(x^-l)(x^^5x-{-6), 

6. 

aj2  +  a^  —  &2  _|_  2  ax, 

(x''-\-3x)(x''-x-6). 

x^..a^^¥-}-2bx, 

(Harvard.) 

x^  -a? -¥-2  ah. 

(Harvard.) 

3. 

x^-f, 

X'  +  y% 

7. 

2x^-x-l^, 

^  +  y% 

3a;2_llaj  +  6, 

a^-hy', 

2:>^-x^-13x-^. 

a^  —  y^. 

(College  Entrance  Board.) 

(College  Entrance  Board.) 

4. 

aj3  4_  aj2  _  2, 

8. 

(tv-vy, 

0^3  +  2  a;2  -  3. 

v^  -  t\ 

(Oornell) 

f  -  v\ 

v'^  —  2vt  4- 1\ 

Pick  out  the  H.C.F.  and  the  L.C.M.  of  the  following: 

9.    ^(x'  +  yyXf  +  zy^m-nyS 

12(x' -^y)'^(t''^zy\m-nY, 
18(m  -  n^fXx"^  +  yyXf"  +  zf\ 

10.   \lax\y  +  zy\y  -  xy\x  +  zf\ 
34aV(?/  +  zy\y  -  xfHx  4-  zy\ 
51a^x\y  +  z)Xx  +  zy\y  -  xf^ 

Reference :    The  chapter  on  H.  C.  F.  and  L.  C.  M.  in  any  algebra. 


20  FRACTIONS 

FRACTIONS 

Define  :  fraction,  terms  of  a  fraction,  reciprocal  of  a  number. 

Look  up  the  law  of  signs  as  it  applies  to  fractions.  Except 
for  this,  fractions  in  algebra  are  treated  exactly  the  same  as 
they  are  in  arithmetic. 

1.  Eeduce  to  lowest  terms  : 

(a)  ^-  (U)  «"-^-  (c)   (S^  +  W-{c  +  dy  , 

W   24'  W  ^4_^4'  («)    («  +  e)^_(6  +  rf).-         C^-^-  ^O 

756  G?  4-  W' 

2.  Eeduce  to  a  mixed  expression  :  (a)   ;  (5)    — ^^^^ — 

11  a  —  h 

3.  Eeduce  to  an  improper  fraction  : 

(a)  451 ;  ih)  Qfi  qt. ;  (c)  a^  -  a&  +  &2 
Add: 


a-\-b 


A      5_i_7iiii5  K  ^ ^^      I  4  —  13  g; 

1     .     1     ^^    1 


x{x  —  a){x—l>)      a{a  —  x)(a  —  'b)      h{h  —  x){h  —  a) 
Multiply : 

8.      ^~y   X  ^^  +  ^^  X  ^'  +  -^^  X  -  • 
a?-\-'if       1)^  -\-^y      6^  +  2/^      c 

Divide : 

11.    (t^y;^^±l-y(t±t^.^±l\.  ^Sheffield.) 


\x^  —  y'^      x^—xyj     \x—y      xy—y 
Simplify : 

\x  x'  J     \  X    ^    x^   J     \         2x-^5yJ 

Reference :    The  chapter  on  Fractions  in  any  algebra. 


COMPLEX   FRACTIONS   AND  FRACTIONAL   EQUATIONS      21 


COMPLEX  FRACTIONS  AND  FRACTIONAL  EQUATIONS 

Define  a  complex  fraction. 
Simplify : 


7     6 


2- 


3    4 
7  '5 


2. 


4.    %-■ 


b'  + 


cb 


6-?  +  ?" 
3     2 


{Harvard.) 


3.    2- 


1- 


'-^ 


5.    Ifm  = 


a-^r 


(1  +  2 


jp: 


a  +  3 


1  —  m      1  — 


•  + 


i> 


1-p 

6.    Simplify  the  expression 

1  1  a^  —  ^ 


\^-\-y- 


x-\-y 
1- 


o:;!/    l-aj^  — 2/^ 


,  what  is  the  value  of 
(Univ,  of  Penn.) 

{Cornell.) 


^  +  y] 


7.    Simplify 


{x-^yy 


1  + 


2x1/ 
{x  -  yy 


1-y 

X 


1+^ 


8.   Solve  ^l+^- 


^Z- 


2y-l 


:7. 


9.    Solve    2l-?(aj2  +  3)=^'  +  l-^'.  . 
3      5  3  o 

10.    How  much  water  must  be  added  to  80  pounds  of  a  5  per 

cent  salt  solution  to  obtain  a  4  per  cent  solution?  (Yale.) 

Reference:    See  Complex  Fractions,  and  the  first  part  of  the 

chapter  on  Fractional  Equations  in  any  algebra. 


22  FRACTIONAL  EQUATIONS 

FRACTIONAL  EQUATIONS 


1.    Solve  for  each  letter  in  turn    -  =  -  +  -. 


2.    Solve  and  check  : 

5  a; +  2     fr,  _  3a?~l^  _  3  aj  + 19      fx-^1 
3 


/o      3a:-l\      3aj  +  19      /aj  +  1  ,  oN 


3.  Solve  and  check: 

4.  Solve  (after  looking  up  the  special  sho7't  method) : 

3a;-l      4  a;- 7^0?       2  a; -3        7  a;- 15 
30  15     ~4      12.^-11  60      * 

5.  Solve  by  the  special  short  method : 

Jl ^^_1 1_ 

x  —  2     x  —  3     X  —  4:     x  —  5' 

6.  At  what  time  between  8  and  9  o'clock  are  the  hands  of  a 
watch  (a)  opposite  each  other  ?  (b)  at  right  angles  ?  (c)  to- 
gether ? 

Work  out  (a)  and  state  the  equations  for  (b)  and  (c). 

7.  The  formula  for  converting  a  temperature  of  F  degrees 
Fahrenheit  into  its  equivalent  temperature  of  C  degrees  Centi- 
grade is  C==^  (F—  32).  Express  F  in  terms  of  C,  and  com- 
pute F  for  the  values  (7=  30  and  (7=  28. 

(College  Entrance  Exam.  Board.) 

8.  What  is  the  price  of  eggs  when  2  less  for  24  cents  raises 
the  price  2  cents  a  dozen  ?  ^  (  Yale.) 

9.  Solve  -^4-     ^  ^ 


-2      4-a;2     a; +  2 

Reference :  The  Chapter  on  Fractional  Equations  in  any  algebra. 
Note  particularly  the  special  sliort  methods,  usu- 
ally given  about  the  middle  of  the  chapter. 


SIMULTANEOUS   EQUATIONS  23 

SIMULTANEOUS  EQUATIONS 

Note.  Up  to  this  point  each  topic  presented  has  reviewed  to  some 
extent  the  preceding  topics.  For  example,  factoring  reviews  the  special 
rules  of  multiplication  and  division;  H.  C.  F.  and  L.  C.  M.  review  factor- 
ing ;  addition  and  subtraction  of  fractions  and  fractional  equations  review 
H.  C.  F.  and  L.  C.  M. ,  etc.  From  this  point  on,  however,  the  interdepend- 
ence is  not  so  marked,  and  miscellaneous  examples  illustrating  the  work 
already  covered  will  be  given  very  frequently  in  order  to  keep  the  whole 
subject  fresh  in  mind. 

1.    Solve  by  three  methods  —  addition  and  subtraction,  substi- 

tution,  and  comparison :  ^  ^        ^ 

\3x-^2y  =  l. 

Solve  and  check:    - 


12E,-llE2  =  b-\-12c, 


r  —  s __25  _r-{- s 
r-f-s  —  9      s  —  r  —  6 


=  0. 


4.  One  half  of  A's  marbles  exceeds  one  half  of  B's  and  C's 
together  by  2  ;  twice  B's  marbles  falls  short  of  A's  and  C's 
together  by  16 ;  if  C  had  four  more  marbles,  he  would  have 
one  fourth  as  many  as  A  and  B  together.  How  many  has 
each  ?  (College  Entrance  Board.) 

5.  The  sides  of  a  triangle  are  a,  b,  c.  Calculate  the  radii  of 
the  three  circles  having  the  vertices  as  centers,  each  being 
tangent  externally  to  the  other  two.  (Harvard.) 


(2x-\-3y  =  7, 


6.  Solve   ■{         ■  graphically ;   then   solve   algebra- 

ix-y  =  l, 

ically  and  compare  results.     (Use  coordinate  or  squared  paper.) 
Factor : 

7.  0^4  +  4.         8.    2di0-1024d.         9.   2(a.«3-l)-7(aj2_l). 

References:    The    chapters    on    Simultaneous    Equations    and 
Graphs  in  any  algebra. 


24  SIMULTANEOUS   EQUATIONS   AND   INVOLUTION 


SIMULTANEOUS  EQUATIONS  AND  INVOLUTION 


f   3 


1.   Solve  < 


-^  =  11 


4:X      Sy 
_3_ 


2J 


Look  up  the  method  of  solv- 
ing vi^hen  the  unknowns  are  in 
the  denominator.     Should  you 


A §_  =  IQi 

4*    clear  of  fractions  ? 


2.    Solve 


1 

y 
1 

z 

1 

X 


1 

z 

X 

1 

y' 


1 

a' 
1 

1 

c 


2x-y  =  4:, 
2x-{-3y  =  12. 

Sx-\-7y.=  5, 
'8x  +  Sy  =  -lS, 


3.  Solve  graphically  and  algebraically 

4.  Solve  graphically  and  algebraically  - 

Review  : 

5.  The  squares  of  the  numbers  from  1  to  25. 

6.  The  cubes  of  the  numbers  from  1  to  12. 

7.  The  fourth  powers  of  the  numbers  from  1  to  5. 

8.  The  fifth  powers  of  the  numbers  from  1  to  3. 

9.  The  binomial  theorem  laws.     (See  Involution.) 
Expand  :  (Indicate  first,  then  reduce.) 

10.  {b-\-yy,  12.    (x'^  +  2ay. 

11.  r^-lj-  13.    (x-y  +  2zy, 

14.  A  train  lost  one  sixth  of  its  passengers  at  the  first  stop, 
25  at  the  second  stop,  20  %  of  the  remainder  at  the  third  stop, 
three  quarters  of  the  remainder  at  the  fourth  stop ;  25  remain. 
What  was  the  original  number  ?  (M.  L  T.) 

References:     The  chapter  on  Involution  in  any  algebra.     Also 
the  references  on  the  preceding  page. 


SQUARE  ROOT  25 

SQUARE  ROOT 

Find  the  square  root  of: 

1.  1  +  16  m^  -  40  m*  -h  10  m  -  8  m^  +  25  m\ 

a"^      6  a      ^  ^       G  X     x^ 
^'  —  H  r  J--L  H  I      .• 

x^       X  a       cv 

3.  Find  the  square  root  to  three  terms  of  x'^  +  5. 

4.  Find  the  square  root  of  337,561. 

5.  Find  the  square  root  of  1823.29. 

6.  Find  to  four  decimal  places  the  square  root  of  1.672. 

(Princeton.) 

7.  Add  -A_  +  -i ^-^. 

{x-Vf      {l-xf     1-x     X 

8.  Find  the  value  of : 

^^'^^-.2x3-g^^7xl+:^I^^-4.0. 
24  14  1-12 

9.  Simplify  \{x  +  y^^  (x  -  yf]  {_{x  +  iff-  {x  -  2/)^] . 

10.  Solve  by  the  short  method : 

_5 2\x-Z      x^Vl      lla?  +  5^Q 

1-x  4  8  16 

11.  It  takes  f  of  a  second  for  a  ball  to  go  from  the  pitcher 
to  the  catcher,  and  i  of  a  second  for  the  catcher  to  handle  it 
and  get  off  a  throw  to  second  base.  It  is  90  feet  from  first 
base  to  second,  and  130  feet  from  the  catcher's  position  to 
second.  A  runner  stealing  second  has  a  start  of  13  feet  when 
the  ball  leaves  the  pitcher's  hand,  and  beats  the  throw  to  the 
base  by  i  of  a  second.  The  next  time  he  tries  it,  he  gets  a 
start  of  only  3^  feet,  and  is  caught  by  6  feet.  What  is  his 
rate  of  running,  and  the  velocity  of  the  catcher's  throw  ? 

{Cornell?) 
Reference :  The  chapter  on  Square  Eoot  in  any  algebra. 


26  THEORY  OF  EXPONENTS 

THEORY  OF  EXPONENTS 

Eeview  the  proofs,  for  positive  integral  exponents,  of : 
I.  a^xa'  =  a'"+^  IV.    Va^  =  a"". 

II.  —z=(f"-".  V.  '^ 


a"  \bj      If 

III.       {a'^y=cf"".  VI.  (abcy  =  a"b"c". 

To  find  the  meaning  of  a  fractional  exponent. 

Assume  that  Law  I  holds  for  all  exponents. 

If  so,  a^  •  a*  •  a^  =  a^  =  al 

Hence,  a^  is  one  of  the  three  equal  factors  (hence  the  cube  root) 

In  the  same  way,  a^  -  a^  -  a""  -  a^  >  a^  =  a^  =  a\ 

4 

Hence,  a'  is  one  of  the  five  equal  factors  (hence  the  fifth  root) 
of  a'-  .■.a^  =  Va\ 

P 

In  the  same  way,  in  general,  a*  =Va^. 

Hence,  the  numerator  of  a  fractional  exponent  indicates  the 
power,  the  denominator  indicates  the  root. 

To  find  the  meaning  of  a  zero  exponent. 

Assume  that  Law  II  holds  for  all  exponents. 

If  SO,  —  =  a"""""  =  a^.     But  by  division,  —  =  1. 

.*.  a^  =  1,  Axiom  I. 

To  find  the  meaning  of  a  negative  exponent. 
Assume  that  Law  I  holds  for  all  exponents. 
If  so,  ^  a"*  X  a"""  =  a"""""  =  a^  =  1. 

Hence,  a'^  X  cr"^  =  1. 

a"* 


THEORY   OF   EXPONENTS  27 

THEORY   OF   EXPONENTS  (Continued) 

Eules : 

To  multiply  quantities  having  the  same  base,  add  exponents. 
To  divide  quantities  having  the  same  base,  subtract  exponents. 
To  raise  a  quantity  to  a  power,  multiply  exponents. 
To  extract  a  root,  divide  the  exponent  of  the  power  by  the  index 
of  the  root. 

1.  Find  the  value  of   3^  -  5  x  4^  +  8~~3  +  1*. 

2.  Find  the  vahie  of   8"^  +  9*  -  2-^  +  1"*  -  7^ 

Give  the  value  of  each  of  the  following: 
3-   ?'  4^  S^  30  X  5,  3  x.5^  30  X  5«,  30  +  5^  30  -  50. 

4.  Express  7^  as  some  power  of  7  divided  by  itself. 

Simplify : 

5.  16*  '  2*  •  32I  (Change  to  the  same  base  first.) 

6     ^[T.  7         (^^)^^^      ■ 

8.    (o;  H-  3  o:*  -  2  cc*)(3  _  2  a;"*  +  4  x'i). 

•    \c^d)       \aW)       VftidA. 


>■  (-^: 


10.   h^      X 


11. 

\</b 


ab-^ 


Reference :  The  chapter  on  Theory  of  Exponents  in  any  algebra. 


28  THEORY   OF   EXPONENTS 

THEORY  OF  EXPONENTS   (Continued) 

Solve  for  x : 

1.   x^  =    4.  2.    x~i  =  8. 

Factor : 

3.  xi  —    9.  5.    aj2a  _  ^-e^ 

4.  xi  +  27.  6.    aM  -  3  a*  +  5  oj*  —  15. 

7.  Find  the  H.  C.  F.  and  L.  C.  M.  of 

a2  +  ah^  H-  ah^  -  b%  a^  -  ah^  -  ah^  -  b\ 

8.  Simplify  the  product  of : 

(ayx'^y,  (hxy-'^y,  and  (y^a~^b~'^y,  (Princeton,) 

9.  Find  the  square  root  of: 

25  ah-'  -  10  ah~i  -49+10  a~h^  -f-  25  a-*6l 


10.  Simplify   yj^^j,' 

11.  Find  the  value  of    ^     -7^    '   ^  +  3^  x  --^ — - —  +  8"3. 

210  (7  a  +  5)<^ 


12.  Express  as  a  power  of  2  :  8^ ;  4^ ;  4^  •  8^  •  16^. 

i_ 

13.  Simplify  ^^^j^^—j     ^        . 

14.  Simplify  {/?¥--^-^- 

.  15.    Expand  (Va  +-^6)^  writing  the  result  with  fractional 
exponents. 

Reference :   The  chapter  on  Theory  of  Exponents  in  any  algebra. 


RADICALS  29 

RADICALS 

1.  Review  all  definitions  in  Radicals,  also  the  methods  of 
transforming  and  simplifying  radicals.  When  is  a  radical  in 
its  simplest  form  ? 

2.  Simplify  (to  simplest  form):    ^-;   yj—;    yjp    3^/-; 


tM  >S^  ("^''-Vt^.'  ^^'  *«'  -'^'- 


'{a  +  by 

3.  Keduce   to   entire    surds:    2V3;    2\/3;    6-^/2;    ay/b^; 

4.  Reduce  to  radicals  of  lower  order  (or  simplify  indices)  : 

</a'',    ^^;     -^27^;     ^ST^V;    ^9^^^^. 

5.  Reduce  to  radicals  of  the  same  degree  (order,  or  index)  : 
V7_  and__-\/ii ;   -^5  and  ^3;   ^7  and   V3;  V^  and   V^; 

-v/c^   Vc^,  and  Vc^ 

6.  Which  is  greater,  V3  or  -v^I?     "v/^S  or  2V2? 

7.  Which  is  greatest,  V3,  ^,  or  -^7?     Give  work  and 
arrange  in  descending  order  of  magnitude. 

Collect : 

8.  Vl28-2V50-f-V72-Vl8. 

9.  2V|  +  iV60-f-Vl5  4-V|. 


10.  V(m  —  nya  +  V(m  +  nya  —  Vam^  -f- ^a(n  —  m)^  —  Va. 

11.  A  and  B  each  shoot  thirty  arrows  at  a  target.  B  makes 
twice  as  many  hits  as  A,  and  A  makes  three  times  as  many 
misses  as  B.     Find  the  number  of  hits  and  misses  of  each. 

(  Lhiiv,  of  Col.) 

Reference :  The  chapter  on  Radicals  in  any  algebra  (first  part 
of  the  chapter). 


30  EADICALS 

RADICALS  (Continued) 

The  most  important  principle  in  Radicals  is  the  following: 
111^  _  _  ____ 

(aby  =  a^b" .     Hence  </ab  =  v^  .  ^6.     Or,  </a  •  ^b=  </ab. 

From  this  also  ^=-^&. 

Va 

Multiply : 

1.  2^4  by  3^/6.  3.    a/2  by  ^4. 

2.  V2by-v/3.  4.   -\/a-\-Vx  by  Va-VS. 

6.  V2  +  V3-V5by  V2-V3  +  V5. 

6     _P,Vi>'-4g,  p      Vp^-4g 

2"^         2  -^  2  2 

Divide : 

7.  V27  by  V3.  9.    "v^  by  V6. 

8.  4Vl8  by  5V32.  10.    V3  by  ^3. 

11.  6  VIM  4- 18  V40  -  45  Vl2  by  3  Vl5.        (Short  division.) 

12.  10Vl8- 4^60+5^100  by  3^30. 
Eationalize  the  denominator : 

13   -1..  JL.  _^.  -1-.   _A- 

V3'     Vt'    2V5'     V^'     7^* 
14  2         .     Va  +  V6.  3 


15. 


V2  +  V3'     -Va-Vb      3-V3 
V3  +  V2 


V6+V3-V2 
Eeview  the  method  of  finding  the  square  root  of  a  binomial 
surd.     (By  inspection  preferably.)     Then  find  square  root  of : 
16.   5  +  2V6.  17.    17-12V2.  18.    7-V33. 

Reference:  The  chapter  on  Eadicals  in  any  algebra,  beginning 
at  Addition  and  Subtraction  of  Eadicals. 


MISCELLANEOUS   EXAMPLES  31 

MISCELLANEOUS  EXAMPLES,  ALGEBRA 
TO  QUADRATICS 

Eesults  by  inspection,  examples  1-10. 
Divide : 

1. 


2. 


ajT7+2/T7 

XTT  -|_  yil 

x  —  y 

i»3  _2/3 

m^  -\-'n? 

m^  -\-n^ 

x  —  y'^ 

Multiply : 

5. 

e-^y- 

6. 

(^-gr-^V- 

7. 

(^,^;-8m)(^a_;-8»)_ 

8. 

(„-...-.-!)■. 

9. 

(3^-4-4r^(3^^-7r«). 

.0. 

(2/-40ir3)(32/*+65  ^3). 

4.      ^  _        ^  _ 

■\/x  —  -\/y^ 

Factor : 

ir.   a;3~64.  13.  h^  —  ^mr^, 

12.   2/^  +  27.  14.  3p-8p*-35. 

Factor,  using  radicals  instead  of  exponents : 

15.    60-7V3^-66.  16.    15m-2Vmn-247i. 

17.  a—h  (factor  as  difference  of  two  squares). 

18.  a—h  (factor  as  difference  of  two  cubes). 

19.  a  —  h  (factor  as  difference  of  two  fourth  powers). 

20.  Find  the  H.  C.  F.  and  L.  C.  M.  of  a:^  +  a;i/*  -  2  ?/,  2  ar^  + 
5  xy'^  +  2  ?/,  2 aj2  —  xy^  —  y, 

x  —  7     x  —  S_x  —  4z     X  —  5 
x—S     x—9     x—5     x—6 

(Princeton,) 


21.    Solve  (short  method) 

c       a       h  ^^ 
22-    Simplify  ^^^_^_^X 

he     ca     ah 


'(a-Jrh+  cf  _  2 
ah  -j-hc-^-ca 


32 


MISCELLANEOUS  EXAMPLES 


MISCELLANEOUS  EXAMPLES, 
ALGEBRA  TO  QUADRATICS   {Continued) 

1.  Solve  fori):     2^-3  =  128. 

2.  Solve  for  ^:     r2=:-27. 

3.  Eind  the  square  root  of  8114.4064.  What,  then,  is  the 
square  root  of  .0081144064  ?  of  811440.64  ?  From  any  of  the 
above  can  you  determine  the  square  root  of  .081144064  ? 

4.  The  H.  C.  F.  of  two  expressions  is  a{a  —  6),  and  their 
L.C.M.  is  a2&(a -f- 6)(a  —  &).  If  one  expression  is  ab(a'^  —  b'^), 
what  is  the  other  ? 

5.  Solve  (short  method)  : 

5     _  2  j:  g;  -  3 
7-x  4 


a;+ll      11  a;  -j-  5  _  ^ 


6.    Solve 


2  _ 
m 


16 


1_1      5^_1^ 

7n      71     p  2 


7.    Simplify  21  Vf  -  5  V|  +  6  V4i-  -  10 V3^  +  ^  Vlli- 


8.   Does  Vl6  X  25  =  4  X  5  ?     Does  Vl6  +  25  =  4  +  5  ? 


9.   Write  the  fraction  - 

4  +  2V3 
and  find  its  value  correct  to  two  decimal  places. 


-with  rational  denominator, 


10.    Simplify 


p+  Vq 


(Princeton.) 


MISCELLANEOUS   EXAMPLES  33 

MISCELLANEOUS   EXAMPLES, 
ALGEBRA  TO  QUADRATICS  (Continued) 

1.  Eationalize  the  denominator  of       _*"_""       _. 

V6-V3  4-3V2 

(Univ.  of  Cal.) 

2.  Simplify  ^^""'"y-  (^^^^*^-  of  Perm.) 

_x 

3.  Find  the  value  of  -^— — — ,  when  aj  =  2.     (Cornell) 

(8  x)i  + 10--2 

4.  Find  the  value  of  a;  if  J       ~  •^  '  (M.  I.  T.) 

5.  A  fisherman  told  a  yarn  about  a  fish  he  had  caught.  If 
the  fish  were  half  as  long  as  he  said  it  was,  it  would  be  10 
inches  more  than  twice  as  long  as  it  is.  If  it  were  4  inches 
longer  than  it  is,  and  he  had  further  exaggerated  its  length 
by  adding  4  inches,  it  would  be  ^  as  long  as  he  now  said  it 
was.  How  long  is  the  fish,  and  how  long  did  he  first  say  it 
was?  (M.LT.) 

6.  The  force  P  necessary  to  lift  a  weight  W  by  means  of  a 
certain  machine  is  given  by  the  formula 

P=a+bW, 
where  a  and  b  are  constants  depending  on  the  amount  of  fric- 
tion in  the  machine.  If  a  force  of  7  pounds  will  raise  a  weight 
of  20  pounds,  and  a  force  of  13  pounds  will  raise  a  weight  of  50 
pounds,  what  force  is  necessary  to  raise  a  weight  of  40  pounds  ? 
(First  determine  the  constants  a  and  b.)  (Harvard,) 

7.  Reduce  to  the  simplest  form :  -vIt^— g*?    — ^—^ ^ ^• 

^  ^"  x^  —  a^ 

8.  Determine  the  H.  C.  F.  and  L.  C.  M.  of  (xy  -  y^  and 
y^  —  x^y.  (College  Entrance  Board.) 

REV.    ALG.  —  3 


34  MISCELLANEOUS   EXAMPLES 

MISCELLANEOUS  EXAMPLES, 
ALGEBRA  TO  QUADRATICS  {Continued) 

1.  Simplify     ^-^^^-2aM. 

a3  _  2  m3 

2.  Simplify,  writing  the  result  with  rational  denominator : 

.1+ iv-r-i.-.*^^ 


"^  ^  '■ 1-.  {M.  I.  T,) 


3.  Find  Vj-VlS. 

4.  Expand  (Va^-VP)^ 

5.  Expand  and  simplify  (1  -2V3  +  3V2)2. 

6.  Solve  the  simultaneous  equations  J  ^  ^  +  ^2/  ^=6? 

I  2  .T-i  -  y-i  =  2. 

(FaZe.) 

7.  Find  to  three  places  of  decimals  the  value  of 


4 


(a  +  &)~'3-       (a^  _  53)- 


(11  a  +  52)6       (ct  _  5)2 

when  a  =  5  and  &  =  3.  (Columbia.) 

8.    Show  that   — ^^^ ^  is  the  negative  of  the  reciprocal  of 

5  +  3  V5 

=-.  (Columbia,') 

5-3V5  ^ 


9.    Solve  and  check  — 1==  =  V3  a;  -f  2  +  V3  a;  —  1. 

V3  a?  +  2       ^ 

10.  Assuming  that  when  an  apple  falls  from  a  tree  the  dis- 
tance {S  meters)  through  which  it  falls  in  any  time  {t  seconds) 
is  given  by  the  formula  S  =  ^yf  (where  ^  =  9.8),  find  to  two 
decimal  places  the  time  taken  by  an  apple  in  falling  15  meters. 

(College  Entrance  Board.) 


MISCELLANEOUS   EXAMPLES  35 

MISCELLANEOUS  EXAMPLES, 
ALGEBRA   TO   QUADRATICS  {Continued) 

Excellent  practice  may  be  obtained  by  solving  the  ordinary 
formulas  used  in  arithmetic,  geometry,  and  physics  orally,  for 
each  letter  in  turn. 

Arithmetic 
p  =  br  a  =  p  -\-  prt 


J  =prt 


Geometry 


K  = 

:\hh 

K  = 

:bJl 

K  = 

4 

K  = 

■•i{b  +  b')h 

K  = 

■■irE' 

C  = 

.2irR 

K  = 

-itRL 

S  = 

=  4  7ri22 

V- 

=  gt 

s- 

=  igt' 

s  - 

2^ 

C-- 

E 
E 

E-. 

2g 

e  - 

bh^m 

E-. 

mv^ 

Physics 


E'H 
R' 


V=i7rR^H 


3 

ttR^E 


KJ    - 

180 

G 
C 

R 
'  R' 

K 

R^ 

K'~ 

R'^ 

tz 

F= 

r 

mil  - 

2^ 

R-. 

V  - 

_  g^ 

4  uH^w 

(7  =  1  (F- 32) 


36  QUADRATIC   EQUATIONS 

QUADRATIC  EQUATIONS 

1.  Define  a  quadratic  equation ;  a  pure  quadratic ;  an  affected 
(or  complete)  quadratic ;  an  equation  in  the  quadratic  form. 

2.  Solve  the  pure  quadratic =  - . 

Eeview  the  first  (or  usual)  method  of  completing  the  square. 
Solve  by  it  the  following: 

3.  x''  +  10x  =  24..  5.    ?il^+^_  =  2i 

2         X—  1 

4.  2  ic2  —  5  a;  =  7.  6.    ax"^  +  bx  +  c  ==  0. 

Eeview  the  solution  by  factoring.     Solve  by  it  the  following : 

7.  a;2  4-8a;4-7  =  0.  9.    3  =  10 a?- 3 a?^. 

8.  24  aj2  =  2  0^  +  15.  10.    -  7  =  6  a;  —  x\ 

Solve,  by  factoring,  these  equations,  which  are  not  quadratics : 
11.    a^  =  16.  12.    a^  =  8.  13.    x^  =  x. 

Review  the  solution  by  formula.     Solve  by  it  the  following: 

14.  5x^-6x  =  S. 

15.  1(0^+1)  _|(2a.-l)  =  -12. 

16.  x'^-{-4:ax  =  12a\ 

17.  3x^  =  2rx-\-2  r\ 

«. 

Solve  graphically: 

18.   x^-2x-S  =  0.  19.    x^^x-2  =  0. 

Reference :   The  chapter  on  Quadratic  Equations  in  any  algebra 
(first  part  of  the  chapter). 


QUADRATIC   EQUATIONS  37 

QUADRATIC  EQUATIONS  (Continued) 

1.    Solve  by  three  methods  —  formula,  factoring,  and  com- 
pleting the  square :  x^  ■j-10x  =  24. 

Eeview  equations  in  the  quadratic  form  and  solve  : 


£i±-^  -1-  6  =  5\/^±^.     (^Let  y  =  a /^^  and  substitute.^ 
X—  l^  ^x  —  3v  ^  X  —  3  / 


X—  3  ^  X  —  3      \ 


5.  3x^-4.x-{-2V3x''-4.x-6  =  21, 

6.  x^  -j-  5x—  5  = 

x^  -^  5x 

Solve  and  check : 

7.  v^+T+ VS'^'^=^=  ^'""^^ 


V3  a;  -  2 


8.    v'a^2-o  +  — ==-5. 
10 1/; 


9.    — VlO  10-^2=     ^  z' 

VlO  z(;  -  9  VlO  w-9 

Give  results  by  inspection : 

10. '  ( Va  +  V&)( Va  -  V6). 

11-   (Vro+  vT9)  (VlO  -  vr9). 

12.  How  many  gallons  each  of  cream  containing  33  % 
butter  fat  and  milk  containing  6  %  butter  fat  must  be  mixed 
to  produce  10  gallons  of  cream  containing  25  %  butter  fat  ? 

13.  I  have  $  G  in  dimes,  quarters,  and  half-dollars,  there  being 
33  coins  in  all.  The  number  of  dimes  and  quarters  together  is 
ten  times  the  number  of  half-dollars.  How  many  coins  of 
each  kind  are  there  ?  (College  Entrance  Board,) 

Reference:   The  last  part  of  the  chapter  on  Quadratic  Equa- 
tions in  any  algebra. 


38  THE   THEORY   OF   QUADRATIC   EQUATIONS 

THE  THEORY  OF  QUADRATIC  EQUATIONS 
I.  To  find  the  sum  and  the  product  of  the  roots. 

The  general  quadratic  equation  is 

ax^-\-bx-\-c  =  0,  (1) 

Or,  x''-{--x-{--  =  0.  (2) 

a        a 

To  derive  the  formula,  we  have  by  transposing 

9  ,    &  c 

a  a 

Completing  the  square, 

x^-\--x-{-( 


a         \2  aj       Aa^      a         4  a- 


Extracting  square  root,  x  +  ——  = 

2  a  2  a 


rn              •                                        b    ,  -\/b^  —  4  ac 
Transposing,  a?  =  -  7—  ± 

2a  2a 


XT  -b±  Vb^  -  4  ac 

Hence,  x  = ^^— 

These  two  values  of  x  we  call  'roots. 

For  convenience  represent  them  by  7\  and  rg. 


Hence, 


Adding,  rj  +  r2  =  -  — -  = (3) 


r2  = 

2a"^ 
b 

V62- 

^  2 

-4 

a 

ac 

V62- 

-4 

ac 

2a 

2 

a 

n 

+  ^2  = 

2b_ 
2a 

a 

THE   THEORY   OF   QUADRATIC   EQUATIONS 


39 


Also, 


Ti 

h 

,  V52-4 

ac 

= 

2a 

b 

2a 

'          2a 

n 

2a 

ac^ 

r.r^ 

52 

b^  —  4:aG 

b"^  —  b^  -\-  4:  ac  __4:  ac  _c 
'         4a2  ~4a2  ""a' 


(4) 


Hence  we  have  shown  that  - 


^1  +  ^2  = , 

a 

and  rir2  =  -  • 
a 


Or,  referring  to  equation  (2)  above,  we  have  the  following  rule  : 

When  the  coefficient  of  x^  is  unity,  the  sum  of  the  roots  is  the 
coefficient  of  x  ivith  the  sign  changed;  the  product  of  the  roots  is 
the  independent  term. 


Examples  : 

1.   a;2-9a;  +  21  =  0. 


r        Sum  of  the  roots  =  9. 
\  Product  of  the  roots  =  21. 


2.  3a;2-7a;-18  =  0. 

3.  -21x  =  ll  -4.x\ 
II.   To  find  the  nature  or  character  of  the  roots. 


Sum  of  the  roots  =  |-. 


[  Product  of  the  roots  =  —  6. 
f        Sum  of  the  roots  =  ^^-. 
\  Product  of  the  roots  =  —  ^-. 


As  before. 


2  a 
b__ 

2a 


ro==- 


V&2- 

-4 

ac 

2 

a 

V&2- 

-4 

ac 

2a 


The  V&2  —  4  ac  determines  the  nature  or  character  of  the 
roots :  hence  it  is  called  the  discriminant. 


40  THE   THEORY   OF   QUADRATIC   EQUATIONS 

If  &2  —  4  ac    is    positive,    the    roots    are    real,    unequal,    and 
either  rational  or  irrational. 

If  &^  —  4  ac  is  negative,  the  roots  are  imaginary  and  unequal. 
If  6^  —  4  ac  is  zero,  the  roots  are  real,  equal,  and  rational. 

Examples  : 

1.    x^-Ax-^2  =  0. 


V&^  —  4  ac  =  VI6  —  8  =  Vs.     .*.  The  roots  are  real,  unequal, 
and  irrational. 

2.    x^  — 4.x +  6  =  0. 


V&^  —  4  ac  =  V16  —  24  =  V—  8.     .*.  The  roots  are  imaginary 
and  unequal. 

3.    x'^-Ax-\-4:  =  0. 

V&^  —  4  ac  =  VI6  —  16  =  VO.     .*.  The  roots  are  real,  equal, 
and  rational. 

III.    To  form  the  quadratic  equation  when  the  roots  are  given. 

Suppose  the  roots  are  3,  —  7. 

x  =  3,  ic  — 3  =  0, 

Then,  \  Or,  ' 

Multiplying  to  get  a  quadratic,  (x  —  3){x  -f-7)  =  0. 

Or,  a;2  +  4  a;  -  21  =  0. 

0}%  use  the  sum  and  product  idea  developed  on  the  preced- 
ing page.     The  coefficient  of  x^  must  be  unity. 

Add  the  roots  and  change  the  sigii  to  get  the  coefficient  of  x. 
Multiply  the  roots  to  get  the  independent  term. 
.  *.  The  equation  is    a^^  +  4  cc  —  21  =  0. 

In  the  same  way,  if  the  roots  are         ^^    ,  '^^^ ,  the  equa- 
tion is 


THEORY   OF   QUADRATIC    EQUATIONS  41 

THE  THEORY  OF  QUADRATIC  EQUATIONS  ^Continued) 

Find  the  sum,  the  product,  and  the  nature  or  character  of 
the  roots  of  the  following : 


2.  9  a?2  _  6  .T  +  1  =  0. 

3.  0^2  +  2  a; +  9734  =  0. 


X  —  '6 

6.  (i»  +  7)(x- 6)=  70. 

7.  x'-x^2  =  3. 


^'  .    ^  +  ^  ~  ^*  8.  pr2  +  gr  +  s  =  0. 


Form  the  equations  whose  roots  are : 

9.   5,  -3.  2±V^"3 

13.  -        . 

14.  I  +  I V37,  I  -  |V37. 
2±V-2 


10. 

2      5 

3?    3- 

11. 

c  4-  d,  c  —  (^. 

12. 

-  3,  -  5. 

15. 


2 


16.  Solve  a;^  —  3  a:  -f  4  =  0.  Check  by  substituting  the 
values  of  x ;  then  check  by  finding  the  sum  and  the  product  of 
the  roots.     Compare  the  amount  of  labor  required  in  each  case. . 

17.  Solve  (x  -  ?>){x  +  2){x'  +  3a?  -  4)=  0. 

18.  Is  e^'  +  2  e^^  +  e^z  ^  2  e^  +  2  +  g-^^  a  perfect  square  ? 

19.  Find  the  square  root  (short  method)  : 

(a;2  -  l)(a;2  _  3  a;  -f  2)(a;2  -  x  -  2). 

„^     ci  1         1.2  a?  — 1.5   ,    .4a?-f  1       .4a;+l 

20.  Solve    -X^  +  :2^^  =  — 5- 

•  21.  The  glass  of  a  mirror  is  18  inches  by  12  inches,  and  it 
has  a  frame  of  uniform  wddth  whose  area  is  equal  to  that  of 
the  glass.     Find  the  width  of  the  frame. 


42 


OUTLINE   OF   SIMULTANEOUS  QUADRATICS 


OUTLINE  OF  SIMULTANEOUS  QUADRATICS 

f  One  equation  linear. 


Simultaneous 
Quadratics 


Case  I. 


Case  II.    ^ 


[The  other  quadratic. 

r2x-\-y  =  7, 
1^2  _p  2^/2  =  22. 

Method  :  Solve  for  x  in  terms  of  2/, 
or  vice  versa,  in  the  linear  and  sub- 
stitute in  the  quadratic. 

Both   equations   homogeneous   and 


1^     of  the  second  degree. 

{ x"^  —  xy  -^  y"^  =  39, 

\2x^-3xy-]-2y''  =  4.3. 

Method  :  Let  y  =  vx,  and  substitute 
in  both  equations. 

Alternate  Method  :  Solve  for  x  in 
terms  of  y  in  one  equation  and  sub- 
titute  in  the  other. 


Case  III. 


x^y 

x"  +2/' 

• 

xy 

Any  two  of  the 

x  —  y 

^give 

quantities 

^^  _l_  ^ 

:k?  —  f 

x^  -\-  xy-\-y'^ 

*. 

.aj2_  xy-^y'^^ 

^x-\-y  =  5, 

\x^  -xy  -{-y'^ 

=  7. 

Method  :  Solve  iov  x  -[-  y  and  x  —y\ 
then  add  to  get  cc,  subtract  to  get  y. 


OUTLINE   OF   SIMULTANEOUS   QUADRATICS 


43 


Simultaneous 

Quadratics 

{Continued) 


Case 
IV. 


Both  equations  symmetrical  or  symmet- 
rical except  for  sign.  Usually  one 
equation  of  high  degree,  the  other  of 
the  first  degree. 

'  x'^-^-y^^  242, 

Method :  Let  x  =  u  -\-  v  and  y  =  u—  v, 
and  substitute  in  both  equations. 

I.    Consider   some   compound    quantity 


Special 
Devices 


like  xy,  -y/x  —  y,  -y/xy,  -  j  ^^c,  as 

the  unknown,  at  first.  Solve  for 
the  compound  unknown,  and 
combine  the  resulting  equation 
with  the  simpler  original  equa- 
tion. 

f  x'^y'^  -{-  a??/  =  6, 

i  oj  +  2  2/  =  -  5. 

II.  Divide   the    equations    member   by 

member.  Then  solve  by  Case  I, 
II,  or  III. 

(:^-y^  =  152, 
Vx-y  =  2. 

III.  Eliminate    the    quadratic    terms. 

Then  solve  by  Case  I,  II,  or  III. 

(xy-\-x  =  15, 
Vxy  -^y  =  l&. 


44  SIMULTANEOUS   QUADRATICS 

SIMULTANEOUS  QUADRATICS 

Solve: 

^      (x-\-y  =  7,  ^      U3x~2y)(2x-3y)=26, 

'    \x^-i-4:xy  =  57.  '     [x-\-l  =  2y. 

2      '2a;2  =  46+2/^  (^x^ -i-Sxy -\-2y''  =  lS, 


xy  +  y^  =  14:.  '     [Sx''-\-2xy-y''  =  3. 

V  +  2/'  =  25,  (x'  +  y'  =  242, 

3.     i  10.      ' 


x  +  y=zl,  [x-\-y  =  2. 


^      :^^  +  2/'  =  2,  ^^      ^x-y+Vx-y  =  6, 


x  —  y  =  2.  '     [xy  =  5. 

a^-^y'  =  28,  ^4.x''- x +y=  67, 


^y  +  xy  —  12  =  0,  fx  —  y  —  -yjx  —  y  =  2, 

x-\-y  =  4..  ^^'  [  0^3-^3^2044.       {Yale.) 

2xy  —  x-\-2y  =  16,  f  x'^ -\- xy  -\- x  =  14, 
14 


Sxy-\-2x-4y  =  10.         '     [y"^  ^  xy -^y  =  2S. 

(Princeton.) 

[2/2  =  4(x  —  2).     Plot  the  graph  of  each  equation. 

[Cornell.) 
^g      {x^^y^  =  xy+37, 

\x-\-y  =  xy  —17.  {Columbia.) 

hi  grouping  the  answers,  he  sure  to  associate  each  value  of 
X  with  the  corresponding  value  of  y. 

17.   The  course  of  a  yacht  is  30  miles  in  length  and  is  in  the 
shape  of  a  right  triangle  one  arm  of  which  is  2  miles  longer 
than  the  other.     What  is  the  distance  along  each  side  ? 
Reference:  The   chapter  on   Simultaneous   Quadratics   in  any 
algebra. 


RATIO   AND   PROPORTION  45 

RATIO   AND  PROPORTION 

1.  Define  ratio,  proportion,  mean  proportional,  third  pro- 
portional, fourth  proportional. 

2.  Find  a  mean  proportional  between  4  and  16 ;  18  and  50 ; 
12  m'^n  and  3  m^i^. 

3.  Find  a  third  proportional  to  4  and  7 ;  5  and  10  ;  a^  _  9 
and  a  —  3. 

4.  Find  a  fourth  proportional  to  2,  5,  and  4 ;  35,  20,  and  14. 

5.  Write   out   the   proofs   for  the  following,   stating   the 
theorem  in  full  in  each  case : 

(a)  The  product  of  the  extremes  equals  etc. 

(6)  If  the  product  of  two  numbers  equals  the  product  of  two 
other  numbers,  either  pair  etc. 

(c)    Alternation.  (e)    Composition. 

{d)  Inversion.  (/)   Division. 

(g)   Composition  and  division. 

(h)  In  a  series  of  equal  ratios,  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  etc. 

(^)    Like  powers  or  like  roots  of  the  terms  of  a  proportion  etc. 

6.  If  a; :  m  : :  13  :  7,  write  all  the  possible  proportions  that 
can  be  derived  from  it.      [See  (5)  above.] 

7.  Given  rs  =  161  m ;  write  the  eight  proportions  that  may- 
be derived  from  it,  and  quote  your  authority. 

8.  (a)  What  theorem  allows  you  to  change  any  proportion 
into  an  equation  ? 

(b)  What  theorem  allows  you  to  change  any  equation  into  a 
proportion  ? 

9.  If  xy=rg,  what  is  the  ratio oi  xtog?  oiytor?  of  y  tog? 

10.  Find  two  numbers  such  that  their  sum,  difference,  and 
the  sum  of  their  squares  are  in  the  ratio  5  :  3  :  51.  (  Yale.) 

Reference :  The  chapter  on  Ratio  and  Proportion  in  any  algebra. 


46  RATIO   AND   PROPORTION 

RATIO  AND  PROPORTION   (Continued) 

An  easy  and  powerful  method  of  proving  four  expressions  in 
proportion  is  illustrated  by  the  following  example : 

Given  a  :b  =  c  :d; 
prove  that  S  a^  -\-  6  ab'^ :  S  a^  -  b  ab^  =  S  c^  +  6  cd^  :S  c^  -  6  cd^ 


Let   «  =  r. 
b 

.-.  a  = 

br. 

Also  -  =  r. 
d 

,\  c  = 

dr. 

Substitute  the  value  of  a  in  the  first  ratio,  and 

.  c  in 

the  second : 

Then    ^«: 
3a3 

+  5  a52 
-5a62 

_  3  &V  +  5  bh^  _ 
3  &3^3  _  5  ^3r 

_  b^r(S  r2 
63r(3  r2 

+  5) 
-6) 

_  3  r2  +  5. 
3r2-5 

Also     3c3  +  5c^2 

3  C3  -  5  C(^2 

_  3  (?3^3  +  5  d¥  _ 
3  d¥3  -  5  d^r 

_  d^r(S  r2 
d3r(3  r2 

+  5) 
-5) 

_  3  r2  +  5. 
3r2-  5 

3a3 

+  5  ab^ 

_  3  c3  +  6  c^2 

3a3 

-6ab^ 

3  c3  -  6  cd2 

Axiom  1. 

Or,  S  a^  +  6  ab^  :S  a^  -  6  ab^  =  S  c^  +  6  cd^iS  c^  ^  6  cd^. 

If  a:b  =  c:  dy  prove : 

1.  a2  +  &2 :  ^2  =  c2  +  d2 :  c\ 

2.  a^  +  3  62 :  a2  -  3  62  =  c2  +  3  (^2 .  (.2  _  3  c^2^ 

3.  a'  +  2b^:2b''  =  ac  +  2bd:2bd. 

4.  2a  +  3c:2a-3c  =  864-12d:86-12d 

5.  a^  -  a6  +  6^ :  ^'  ~  ^'  =0^  -  cd  +  ^2 .  gL:^' . 

a  c 

6.  The  second  of  three  numbers  is  a  mean  proportional 
between  the  other  two.  The  third  number  exceeds  the  sum  of 
the  other  two  by  20 ;  and  the  sum  of  the  first  and  third  exceeds 
three  times  the  second  by  4.     Find  the  numbers. 

7.  Three  numbers  are  proportional  to  5,  7,  and  9 ;  and  their 
sum  is  14.     Find  the  numbers.  (College  Entrance  Board.) 

8.  A  triangular  field  has  the  sides  15,  18,  and  27  rods, 
respectively.  Find  the  dimensions  of  a  similar  field  having 
4  times  the  area. 


ARITHMETICAL   PROGRESSION  47 

ARITHMETICAL   PROGRESSION 

1.  Define  an  arithmetical  progression. 
Learn  to  derive  the  three  formulas  in  arithmetical  progression: 

1  =  a-\-(7i  —  l)d, 

2.  Find  the  sum  of  the  first  50  odd  numbers. 

3.  In  the  series  2,  5,  8,  •••,  which  term  is  92  ? 

4.  How  many  terms  must  be  taken  from  the  series  3,  5,  7, 
•••,  to  make  a  total  of  255  ? 

5.  Insert  5  arithmetical  means  between  11  and  32. 
6    Insert  9  arithmetical  means  between  7|-  and  30. 

7.  Find  x,  if  3 -{-2  x,  5 -\-6x,  9 +  5  x  are  in  A.  P. 

8.  The  7th  term  of  an  arithmetical  progression  is  17,  and 
the  13th  term  is  59.     Find  the  4th  term. 

9.  How  can  you  turn  an  A.  P.  into  an  equation  ? 

10.  Given  a  =  —  f,  n  =  20,  8  =  —  ^,  find  d  and  I. 

11.  Find  the  sum  of  the  first  n  odd  numbers. 

12.  An  arithmetical  progression  consists  of  21  terms.  The 
sum  of  the  three  terms  in  the  middle  is  129 ;  the  sum  of  the 
last  three  terms  is  237.  Find  the  series.  (Look  up  the  short 
method  for  such  problems.)  (Mass.  Inst,  of  Technology.) 

13.  B  travels  3  miles  the  first  day,  7  miles  the  second  day, 
11  miles  the  third  day,  etc.  In  how  many  days  will  B  over- 
take A  who  started  from  the  same  point  8  days  in  advance  and 
who  travels  uniformly  15  miles  a  day  ? 

Reference:     The  chapter  on  Arithmetical  Progression  in   any 
algebra. 


48  GEOMETRICAL   PROGRESSION 

GEOMETRICAL  PROGRESSION 

1.    Define  a  geometrical  progression. 
Learn  to  derive  the  four  formulas" in  geometrical  progression : 

rl  —  a 


I.     /  =  ar""\ 
11.    ^^ar^-a 


III.  S  =  - 

r-1 

IV.  ^OD=^ 


1  —  r 


r-1 

2.  How  many  terms  must  be  taken  from  the  series  9,  18, 
36,  --to  make  a  total  of  567  ? 

3.  In  the  a.  P.  2,  6,  18,  ...,  which  term  is  486? 

4.  Find  Xy  if  2  x  —  4,  5  a;  —  7,  10  oj  +  4  are  in  geometrical 
progression. 

5.  How  can  you  turn  a  G.  P.  into  an  equation  ? 

6.  Insert  4  geometrical  means  between  4  and  972. 

7.  Insert  6  geometrical  means  between  ^^  and  5120. 

8.  Given  a  =  —  2,  n  ^  5,  Z  =  —  32 ;  find  r  and  S. 

9.  If  the  first  term  of  a  geometrical  progression  is  12  and 
the  sum  to  infinity  is  36,  find  the  4th  term. 

10.  If  the  series  3^,  2^  •••  be  an  A.  P.,  find  the  97th  term. 
If  a  G.  P.,  find  the  sum  to  infinity. 

11.  The  third  term  of  a  geometrical  progression  is  36;  the 
6th  term  is  972.     Find  the  first  and  second  terms. 

12.  Insert  between  6  and  16  two  numbers,  such  that  the 
first  three  of  the  four  shall  be  in  arithmetical  progression,  and 
the  last  three  in  geometrical  progression. 

13.  A  rubber  ball  falls  from  a  height  of  40  inches  and  on 
each  rebound  rises  40  %  of  the  previous  height.  Find  by 
formula  how  far  it  falls  on  its  eighth  descent.  (Yale.) 

Reference :     The  chapter  on  Geometrical  Progression  in  any 
algebra. 


THE   BINOMIAL   THEOREM  49 

THE   BINOMIAL  THEOREM 

1.  Eeview  the  Binomial  Theorem  laws.     (See  Involution.) 
Expand : 

2.  (h-ny.  5.    i-^-^'xy 

3.  {x^x-y,  6.    (a;2-aj  +  2)3. 

V^    ay  *  '  \  y        &'  / 

8.  (fl  +  by  =:or  -^  na"-^h  +  ^^^  ~J'^a'-^Ij^ 

1  *  <o 

n(n  -  l)(n  -  2)      3         „(„  _  !)(„  _  2)(n  -  3)      ^m  ,  ... 
^        1-2-3  1-2. 3-4  ^" 

Show  by  observation  that  the  formula  for  the 

(r  +  l)th  term  =  /^(^L^AKg  -  ^)  '"  (n  -  r±l)^n-r^, 

9.  Indicate  what  the  97th  term  of  (a  +  by  would  be. 

10.  Using  the  expansion  of  (a  +  by  in  (8),  derive  a  formula 
for  the  rth  term  by  observing  how  each  term  is  made  up,  then 
generalizing. 

Using  either  the  formula  in  (8)  or  (10),  whichever  you  are 
familiar  with,  find : 

/  1  \  30 

11.  The  4th  term  ofla-\--\  . 

12.  The  8th  term  of  (1  +  xVyY^. 

13.  The  jniddle  term  of  (2  a*  -y  \/ay^ 

2X12 
xj 

15.    The  term  containing  x^^  in  \  'y?- )  . 

V         ^J 
Reference:  The  chapter  on  The  Binomial  Theorem  in  any  algebra. 

REV.    ALG. — 4 


14.    The  term  not  containing  x  in  [qi?  — 


50  MISCELLANEOUS   EXAMPLES 

MISCELLANEOUS  EXAMPLES,  QUADRATICS  AND  BEYOND 

1.  Solve   the  equation   oi?  —  1.6  a?  —  .23  =  0,  obtaining  the 
values  of  the  roots  correct  to  three  significant  figures. 

{Harvard^ 

2.  Write  the  roots  of  (x^  -\-2  x){x'  -2  x  -?>){x^  -  x^V)  =^, 

{Sheffield  Scientific  School.) 

3.  Solve  2V2¥+2  +  V2¥+l  =  ^^  ^  +  4^  (Yale^ 

TT  

4.  Solve  the  equation  F=  —  (B  -\-x-\-  VBx)  for  x,  taking 

o 

11=  6,  J5  =  8,  and  V=  2S ;  and  verify  your  result.    (Harvard,) 

Solve  {^^2/  =  2:3, 

V-f-2/2  =  5(a^  +  2/)  +  2. 


6.  Solve2a;2_4a;+3Vaj2-2a;-h6  =  15.     (Coll.  Ent.  Board.) 

7.  Find  all  values  of  x  and  ?/  which  satisfy  the  equations : 

r  Va;  +  V?/  =  4, 

:  y.  (Mass.  Inst,  of  Technology.) 


-Vx  H-  1  —  'Vx      -Vx  +  1  4-  Va? 

8.  If  ot  and  yS  represent  the  roots  of  px'^  -^qx  -\-  r  =  Oy  find 
a  +  p,  a  —  /3,  and  a/?  in  terms  of  p,  q,  and  ?\  (Princeton.) 

9.  Form    the   equation  whose    roots   are    2H-V— 3   and 

10.  Determine,  without  solving,  the  character  of  the  roots  of 

9  x^  —24  x-\-16  =0.     (College  Entrance  Board.) 

11.  li  a:b  =  c:d,  prove  that  a-{-b:  c-\-d  =  Va^^b'^ :  Vc^-{-d\ 

(College  Entrance  Boai'd.) 

12.  Givena:6  =  c:d   Prove  that a^H-^^ .- -^  =  02  +  ^2:-  ^' 


a-\-l)  c-\-d 

(Sheffield.) 

13.    The  9th  term  of  an  arithmetical  progression  is  \  ;  the 
16th  term  is  f.     Find  the  first  term.  (Regents.) 


MISCELLANEOUS   EXAMPLES  51 

MISCELLANEOUS  EXAMPLES,  QUADRATICS  AND  BEYOND 

(Continued) 

Solve  graphically : 

1.   x^-x-6  =  0.  2.   aj2_,_3^_io^O. 

3.  Find  four  numbers  in  arithmetical  progression,  such 
that  the  sum  of  the  first  two  is  1,  and  the  sum  of  the  last  two 
is  - 19. 

4.  What  number  added  to  2,  20,  9,  34,  will  make  the 
results  proportional  ? 

/  hi 

5.  Find  the  middle  term  of  (  3  a^  +  ^ 

V         2 

6.  Solve     -^±i-  =  ?-5jIl|_l ^ (Princeton.) 

7.  A  strip  of  carpet  one  half  inch  thick  and  29|-  feet  long 
is  rolled  on  a  roller  four  inches  in  diameter.  Find  how  many- 
turns  there  will  be,  remembering  that  each  turn  increases  the 
diameter  by  one  inch,  and  that  the  circumference  of  a  circle 
equals  (approximately)  ^-^-  times  the  diameter.  (Harvard.) 

8.  The  sum  of  the  first  three  terms  of  a  geometrical  progres- 
sion is  21,  and  the  sum  of  their  squares  is  189.  What  is  the 
first  term  ?  (  Yale.) 

9.  Find  the  geometrical  progression  whose  sum  to  infinity 
is  4,  and  whose  second  term  is  f . 


10.  Solve     4.x  +  W3x''-7  x  +  S  =  3x'^-3x  +  6. 

11.  Solve   p.^  +  3^^-5^-4, 

1  2xy  +  3y'  =  -S. 

12.  Two  hundred  stones  are  placed  on  the  ground  3  feet 
apart,  the  first  being  3  feet  from  a  basket.  If  the  basket  and 
all  the  stones  are  in  a  straight  line,  how  far  does  a  person 
travel  who  starts  from  the  basket  and  brings  the  stones  to 
it  one  by  one  ? 


52  MISCELLANEOUS   EXAMPLES 

MISCELLANEOUS  EXAMPLES,  QUADRATICS  AND   BEYOND 

{Continued) 

Solve  graphically  ;  and  check  by  solving  algebraically  : 

^x-\-y  =  1. 
2.   x2- 3  0.^-18  =  0.  3.    .t2^  3  a? -10  =  0. 

Determine  the  value  of  m  for  which  the  roots  of  the  equa- 
tion will  be  equal :  (Hint  :  See  page  40.  To  have  the  roots  equal, 
6'^  —  4  ac  must  equal  0.) 

4.  2  0^2  _  ^^^y,  _|.  12 1  =  0. 

5.  (m  —  l)i^2  +  mo?  H- 2  ??i  —  3  —  0. 

6.  If  2  a  -I-  3  &  is  a  root  of  ^^^  _  5  ^^,  _  4  ^2  _^  9  ^2  ^  q^ 
find  the  other  root  without  solving  the  equation. 

(Univ.  of  Penn») 

7.  How   many   times    does    a    common    clock    strike    in 

12  hours  ? 

2        1         1 


8.    Find  the  sum  to  infinity  of 


— > 


V2     V2    2V2 


e.    Solve  (f+^y-6(|  +  ^-)+8  =  0. 

10.  Find  the  value  of  the  recurring  decimal  2.214214  —. 

11.  A  man  purchases  a  $500  piano  by  paying  monthly 
installments  of  $  10  and  interest  on  the  debt.  If  the  yearly 
rate  is  6%,  what  is  the  total  amount  of  interest? 

12.  The  arithmetical  mean  between  two  numbers  is  42^,  and 
their  geometrical  mean  is  42.     Find  the  numbers. 

(College  Entrance  Exam.  Board.) 

13.  If  the  middle  term  oi  fSx =.  )  is  equal  to  the  fourth 

V  2VxJ 

term  of  f  2Va^H ) ,  find  the  value  of  x.  (M.  L  T.) 

2  xj 


PROBLExMS  53 

PROBLEMS 
Linear  Equations,  One  Unknown 

1.  A  train  running  30  miles  an  hour  requires  21  minutes 
longer  to  go  a  certain  distance  than  does  a  train  running 
36  miles  an  hour.     How  great  is  the  distance?  (Oornell.) 

2.  A  man  can  walk  2|  miles  an  hour  up  hill  and  3^  miles  an 
hour  down  hill.  He  walks  56  miles  in  20  hours  on  a  road  no 
part  of  which  is  level.     How  much  of  it  is  up  hill  ?       (Yale.) 

3.  A  physician  having  100  cubic  centimeters  of  a  6  %  solu- 
tion of  a  certain  medicine  wishes  to  dilute  it  to  a  3^  %  solution. 
How  much  water  must  he  add  ?  (A  6  %  solution  contains  6  % 
of  medicine  and  94  %  of  water.)  (Case.) 

4.  A  clerk  earned  $501  in  a  certain  number  of  months.  His 
salary  was  increased  25  %,  and  he  then  earned  $450  in  two 
months  less  time  than  it  had  previously  taken  him  to  earn 
$504.     What  was  his  original  salary  per  month  ? 

(College  Entrance  Board.) 

6.  A  person  who  possesses  $15,000  employs  a  part  of  the 
money  in  building  a  house.  He  invests  one  third  of  the  money 
which  remains  at  6  %,  and  the  other  two  thirds  at  9  %,  and 
from  these  investments  he  obtains  an  annual  income  of  $500. 
What  was  the  cost  of  the  house  ?  (M.  I.  T.) 

6.  Two  travelers  have  together  400  pounds  of  baggage.  One 
pays  $1.20  and  the  other  $1.80  for  excess  above  the  weight 
carried  free.  If  all  had  belonged  to  one  person,  he  would  have 
had  to  pay  $4.50.     How  much  baggage  is  allowed  to  go  free? 

(Yale.) 

7.  A  man  who  can  row  4^  miles  an  hour  in  still  water  rows 
downstream  and  returns.  The  rate  of  the  current  is  2\  miles 
per  hour,  and  the  time  required  for  the  trip  is  13  hours.  How 
many  hours  does  he  require  to  return  ? 


54  PROBLEMS 

Simultaneous  Equations,  Two  and  Three  Unknowns 

1.  A  manual  training  student  in  making  a  bookcase  finds 
that  the  distance  from  the  top  of  the  lowest  shelf  to  the  under 
side  of  the  top  shelf  is  4  ft.  6  in.  He  desires  to  put  between 
these  four  other  shelves  of  inch  boards  in  such  a  way  that  the 
book  space  will  diminish  one  inch  for  each  shelf  from  the  bot- 
tom to  the  top.  What  will  be  the  several  spaces  between 
the  shelves? 

2.  A  quantity  of  water,  sufficient  to  fill  three  jars  of  differ- 
ent sizes,  will  fill  the  smallest  jar  4  times,  or  the  largest  jar 
twice  with  4  gallons  to  spare,  or  the  second  jar  three  times  with 
2  gallons  to  spare.     What  is  the  capacity  of  each  jar?   (Case,) 

3.  A  policeman  is  chasing  a  pickpocket.  When  the  police- 
man is  80  yards  behind  him,  the  pickpocket  turns  up  an  alley; 
but  coming  to  the  end,  he  finds  there  is  no  outlet,  turns  back, 
and  is  caught  just  as  he  comes  out  of  the  alley.  If  he  had  dis- 
covered that  the  alley  had  no  outlet  when  he  had  run  halfway 
up  and  had  then  turned  back,  the  policeman  would  have  had  to 
pursue  the  thief  120  yards  beyond  the  alley  before  catching 
him.     How  long  is  the  alley  ?  (Harvard.) 

4.  A  and  B  together  can  do  a  piece  of  work  in  14  days. 
After  they  have  worked  6  days  on  it,  they  are  joined  by  C  who 
works  twice  as  fast  as  A.  The  three  finish  the  work  in  4 
days.     How  long  would  it  take  each  man  alone  to  do  it  ? 

(Columbia.) 

5.  In  a  certain  mill  some  of  the  .workmen  receive  $  1.50  a 
day,  others  more.  The  total  paid  in  wages  each  day  is  $  350. 
An  assessment  made  by  a  labor  union  to  raise  $  200  requires 
$1.00  from  each  man  receiving  $1.50  a  day,  and  half  of  one 
day's  pay  from  every  man  receiving  more.  How  many  men 
receive  $  1.50  a  day  ?  (Harvard.) 


PROBLEMS  56 

6.  There  are  two  alloys  of  silver  and  copper,  of  which  one 
contains  twice  as  much  copper  as  silver,  and  the  other  three 
times  as  much  silver  as  copper.  How  much  must  be  taken 
from  each  to  obtain  a  kilogram  of  an  alloy  to  contain  equal 
quantities  of  silver  and  copper  ?  {M.  I.  T.) 

7.  Two  automobiles  travel  toward  each  other  over  a  distance 
of  120  miles.  A  leaves  at  9  a.m.,  1  hour  before  B  starts  to 
meet  him,  and  they  meet  at  12  :  00  m.  If  each  had  started  at 
9  :  15  A.M.,  they  would  have  met  at  12 :  00  m.  also.  Find  the 
rate  at  which  each  traveled.  {M.  I.  T.) 

Quadratic  Equations 

1.  Telegraph  poles  are  set  at  equal  distances  apart.  In 
order  to  have  two  less  to  the  mile,  it  will  be  necessary  to  set 
them  20  feet  farther  apart.     Find  how  far  apart  they  are  now. 

{Yale.) 

2.  The  distance  S  that  a  body  falls  from  rest  in  t  seconds 
is  given  by  the  formula  aS  =  16  f^.  A  man  drops  a  stone  into 
a  well  and  hears  the  splash  after  3  seconds.  If  the  velocity 
of  sound  in  air  is  1086  feet  a  second,  what  is  the  depth  of  the 
well?  (Yale.) 

3.  It  requires  2000  square  tiles  of  a  certain  size  to  pave  a 
hall,  or  3125  square  tiles  whose  dimensions  are  one  inch  less. 
Find  the  area  of  the  hall.  How  many  solutions  has  the  equa- 
tion of  this  problem  ?  How  many  has  the  problem  itself  ? 
Explain  the  apparent  discrepancy.  {Cornell.) 

4.  A  rectangular  tract  of  land,  800  feet  long  by  600  feet 
broad,  is  divided  into  four  rectangular  blocks  by  two  streets  of 
equal  width  running  through  it  at  right  angles.  Find  the 
width  of  the  streets,  if  together  they  cover  an  area  of  77,500 
square  feet.  {M.  I.  T.) 


56  PROBLEMS 

5.  (a)  The  height  y  to  which  a  ball  thrown  .vertically  upward 
with  a  velocity  of  100  feet  per  second  rises  in  x  seconds  is 
given  by  the  formula,  y  =  100  x—  16  x^.  In  how  many  seconds 
will  the  ball  rise  to  a  height  of  144  feet  ? 

(h)  Draw  the  graph  of  the  equation  y  =  100  x  —  lQ>  x'^. 

.  (College  Entrance  Board.) 

6.  Two  launches  race  over  a  course  of  12  miles.  The  first 
steams  7^  miles  an  hour.  The  other  has  a  start  of  10  minutes, 
runs  over  the  first  half  of  the  course  with  a  certain  speed,  but 
increases  its  speed  over  the  second  half  of  the  course  by  2 
miles  per  hour,  winning  the  race  by  a  minute.  What  is  the 
speed  of  the  second  launch  ?  Explain  the  meaning  of  the 
negative  answer.  (Sheffield  ScientifiG  School.) 

7.  The  circumference  of  a  rear  wheel  of  a  certain  wagon  is 
3  feet  more  than  the  circumference  of  a  front  wheel.  The 
rear  wheel  performs  100  fewer  revolutions  than  the  front 
wheel  in  traveling  a  distance  of  6000  feet.  How  large  are  the 
wheels  ?  (Harvard.) 

8.  A  man  starts  from  home  to  catch  a  train,  walking  at  the 
rate  of  1  yard  in  1  second,  and  arrives  2  minutes  late.  If  he 
had  walked  at  the  rate  of  4  yards  in  3  seconds,  he  would  have 
arrived  2i  minutes  early.  Find  the  distance  from  his  home  to 
the  station.  (College  Entrance  Board.) 

Simultaneous  Quadratics 

1.  Two  cubical  coal  bins  together  hold  280  cubic  feet  of  coal, 
and  the  sum  of  their  lengths  is  10^  feet.  Find  the  length  of 
each  bin. 

2.  The  sum  of  the  radii  of  two  circles  is  25  inches,  and  the 
difference  of  their  areas  is  125  tt  square  inches.  Find  the 
radii. 


PROBLEMS  •  57 

3.  The  area  of  a  right  triangle  is  150  square  feet,  and  its 
hypotenuse  is  25  feet.     Find  the  arras  of  the  triangle. 

4.  The  combined  capacity  of  two  cubical  tanks  is  637  cubic 
feet,  and  the  sura  of  an  edge  of  one  and  an  edge  of  the  other 
is  13  feet,  (cx)  Find  the  length  of  a  diagonal  of  any  face  of 
each  cube,  (h)  Find  the  distance  from  upper  left-hand  corner 
to  lower  right-hand  corner  in  either  cube. 

5.  A  and  B  run  a  raile.  In  the  first  heat  A  gives  B  a  start 
of  20  yards  and  beats  hira  by  30  seconds.  In  the  second  heat 
A  gives  B  a  start  of  32  seconds  and  beats  hira  by  9j-\  yards. 
Find  the  rate  at  which  each  runs.  {Sheffield.) 

6.  After  street  improvement  it  is  found  that  a  certain  corner 
rectangular  lot  has  lost  -^-^  of  its  length  and  -^^  of  its  width. 
Its  perimeter  has  been  decreased  by  28  feet,  and  the  new  area 
is  3024  square  feet.     Find  the  reduced  diraensions  of  the  lot. 

(College  Entrance  Board) 

7.  A  man  spends  $  539  for  sheep.  He  keeps  14  of  the  flock 
that  he  buys,  and  sells  the  remainder  at  an  advance  of  $2 
per  head,  gaining  %  28  by  the  transaction.  How  many  sheep 
did  he  buy,  and  what  was  the  cost  of  each  ?  (  Yale.) 

8.  A  boat's  crew,  rowing  at  half  their  usual  speed,  row  3 
miles  downstream  and  back  again  in  2  hours  and  40  minutes. 
At  full  speed  they  can  go  over  the  same  course  in  1  hour  and 
4  minutes.  Find  the  rate  of  the  crew,  and  the  rate  of  the  cur- 
rent in  miles  per  hour.  {College  Entrance  Board.) 

9.  Find  the  sides  of  a  rectangle  whose  area  is  unchanged  if 
its  length  is  increased  by  4  feet  and  its  breadth  decreased  by 
3  feet,  but  which  loses  one  third  of  its  area  if  the  length  is 
increased  by  16  feet  and  the  breadth  decreased  by  10  feet. 

(M.  I.  T.) 


COLLEGE  ENTRANCE  EXAMINATIONS 
UNIVERSITY  OF  CALIFORNIA 


ELEMENTARY   ALGEBRA 


1.  If  a  =  4,  &  =  —  3,  c  =  2,  and  d  =  —  4t,  find  the  value  of : 

(a)  aW  -  3  cc?2  _^  2(3  a  -  h)(c  -  2  d), 
(6)  2  a^  -  3  6^  +  (4  c^  +  c?3)(4  c^  +  d"), 

2.  Reduce  to  a  mixed  number : 

3  g^  -  4  g^  -  10  g^  +  41  g  -  28 
g2  -  3  g  +  4 
Simplify : 

g+2  6-2 


g2  +  3  g  —  40      g&  -  5  &  +  3  g  - 15 

/        2  -  3  6  -  2c\     g2-4c2  4-9&2  +  6g& 


V  g  +  2        J  2g2  +  g-6 

5.  A's  age  10  years  hence  will  be  4  times  what  B^s  age  was 
11  years  ago,  and  the  amount  that  A's  age  exceeds  B\s  age  is 
one  third  of  the  sum  of  their  ages  8  years  ago.  Find  their 
present  ages. 

6.  Draw  the  lines  represented  by  the  equations 

3  aj ~ 2  2/  =  13  and  2  x-[-by  =  — 4., 

and  find  by  algebra  the  coordinates  of  the  point  where  they 
intersect. 

o.  -.       .1  . .  {hx  —  ay ^ly^  —  ah, 

7.  Solve  the  equations     \  70/       ox 

^  [^     ^_^=3  2(a;  — 2g). 

8.  Solve  {2  x-\-l){3  x-2)-{5  X  -l){x  -2)  =  41. 

58 


COLLEGE  ENTRANCE  EXAMINATIONS        59 

COLORADO   SCHOOL   OF  MINES 


ELEMENTARY  ALGEBRA 


1.  Solve  by  factoring :  ot^  -]- 30  x  =  11  x\ 

2.  Show  that  1  _fa'  +  b''-c''V 

=  (a  +  6  +  G)(a  +  &  -  c){a  -  &  4-  c)(b  +  c  -  a)  ^4.o?h\ 

3.  How  many  pairs  of  numbers  will  satisfy  simultaneously 
the  two  equations 

x-^y  =  3? 

Show  by  means  of  a  graph  that  your  answer  is  correct. 
What  is  meant  by  eliminating  x  in  the  above  equations  by 
substitution?  by  comparison?  by  subtraction? 

4.  Find  the  square  root  of  223,728. 

5.  Simplify:  (a)   V|+Vl2-Vf. 

(P)  (_V_3V^4)4. 

6.  Solve  the  equation 

.03  x"  -  2.23  X  + 1.1075  =  0. 

7.  How  far  must  a  boy  run  in  a  potato  race  if  there  are  n 
potatoes  in  a  straight  line  at  a  distance  d  feet  apart,  the  first 
being  at  a  distance  a  feet  from  the  basket? 


60        COLLEGE  ENTRANCE  EXAMINATIONS 

COLUMBIA   UNIVERSITY 


ELEMENTARY  ALGEBEA   COMPLETE 


Time  :   Three  Hours 

'  Six  questions  are  required ;  two  from  Group  A,  two  from  Group  B, 
and  both  questions  of  Group  C.  No  extra  credit  will  be  given  for  more 
than  six  questions. 


Group  A 

(a)  Eesolve  the  following  into  their  prime  factors : 

(1)  {x^-fy-y^. 

(2)  10x^-1  x-Q, 

(b)  Find  the  H.  C.  F.  and  the  L.  C.  M.  of 
x^-3x^-j-x-3, 
a^-Sa^-x-i-S. 


2.  (a)  Simplify 

y    ^    "  .y    X 
i_^i    "^    i_i 

X      y  x      y 

(h)  If  x'.y  =  {x  —  zy :  {y  —  zf,  prove  that  a;  is  a  mean  pro- 
portional between  x  and  y. 

3.  A  crew  can  row  10  miles  in  50  minutes  downstream,  and 
12  miles  in  an  hour  and  a  half  upstream.  Find  the  rate  of 
the  current  and  of  the  crew  in  still  water. 


COLLEGE   ENTRANCE   EXAMINATIONS  61 

COLUMBIA   UNIVERSITY   {Continued) 

Group  B 

4.  (a)  Determine  the  values  of  k  so  that  the  equation 

(2  +  k)x''  4-  2  A;aj  +  1  =  0 
shall  have  equal  roots. 

(h)  Solve  the  equations 

2x--3tj  =  0. 

(c)  Plot  the  following  two  equations,  and  find  from  the 
graphs  the  approximate  values  of  their  common  solutions : 

x'-\-f  =  25, 
Ax'-\-9y'  =  lU, 

5.  Two  integers  are  in  the  ratio  4  :  5.  Increase  each  by  15, 
and  the  difference  of  their  squares  is  999.  What  are  the 
integers  ? 

6.  A  man  has  $539  to  spend  for  sheep.  He  wishes  to  keep 
14  of  the  flock  that  he  buys,  but  to  sell  the  remainder  at  a 
gain  of  $2  per  head.  This  he  does  and  gains  $28.  How 
many  sheep  did  he  buy,  and  at  what  price  each  ? 

Group  C 

7.  (a)  Find  the  seventh  term  of  [  a  +  -Vl 

(b)  Derive  the  formula  for  the  sum  of  n  terms  of  an  arith- 
metic progression. 

8.  A  ball  falling  from  a  height  of  60  feet  rebounds  after 
each  fall  one  third  of  its  last  descent.  What  distance  has 
it  passed  over  when  it  strikes  the  ground  for  the  eighth  time  ? 


62  COLLEGE   ENTRANCE   EXAMINATIONS 

CORNELL   UNIVERSITY 


ELEMENTARY   ALGEBRA 


1.  Find  the  H.C.F: 

a^  —  xy^  +  x^y  —  y^, 
x^-^2  xy  -  3  y\ 

2.  Solve  the  following  set  of  equations : 

xi-y  =  -l, 
x-\-3y-^2z  =  -4., 
X  —  y  -{-4:Z  =  5. 

3.  Expand  and  simplify  : 


(-4)' 


4.  An  automobile  goes  80  miles  and  back  in  9  hours.  The 
rate  of  speed  returning  was  4  miles  per  hour  faster  than  the 
rate  going.     Find  the  rate  each  way. 

5.  Simplify : 

"''''Mm 


aj-f-lV     fx  —  1 


x  —  lj      \x-\-l 


6.    Solve  for  x : 


l^±l-6  =  . 


x''-{-2x-S 


7.  A,  B,  and  C,  all  working  together,  can  do  a  piece  of 
work  in  2|  days.  A  works  twice  as  fast  as  C,  and  A  and  C 
together  could  do  the  work  in  4  days.  How  long  would  it 
take  each  one  of  the  three  to  do  the  work  alone  ? 


COLLEGE   ENTRANCE   EXAMINATIONS  63 

CORNELL   UNIVERSITY 


INTEEMEDIATE   ALGEBRA 


1.    Solve  the  following  set  of  equations  : 


2.  Simplify:         (a)V6-V20.  (h)       1+V^_±i 

iJ^^x'  +  l  +  x" 

3.  Find,  and  simplify,  the  23d  term  in  the  expansion  of 

f2_^  _  3  Y^ 
3        4J  • 

4.  The  weight  of  an  object  varies  directly  as  its  distance 
from  the  center  of  the  earth  when  it  is  below  the  earth's  sur- 
face, and  inversely  as  the  square  of  its  distance  from  the  center 
when  it  is  above  the  surface.  If  an  object  weighs  10  pounds  at 
the  surface,  how  far  above,  and  how  far  below  the  surface  will 
it  weigh  9  pounds  ?  (The  radius  of  the  earth  may  be  taken  as 
4000  miles.) 

5.  Solve  the  following  pair  of  equations  for  x  and  y : 

a;2  _|_  ^2  ^  4, 

a;  =  (l+V2)2/-2. 

6.  Find  the  value  of  — 1+A-^ — ,  when  x  =  2. 

(8^)* +  10^-' 

7.  From  a  square  of  pasteboard,  12  inches  on  a  side,  square 
corners  are  cut,  and  the  sides  are  turned  up  to  form  a  rectan- 
gular box.  If  the  squares  cut  out  from  the  corners  had  been 
1  inch  larger  on  a  side,  the  volume  of  the  box  would  have 
been  increased  28  cubic  inches.  What  is  the  size  of  the  square 
corners  cut  out  ?     (See  the  figure  on  the  blackboard.) 


64  COLLEGE   ENTRANCE   EXAMINATIONS 

HARVARD  UNIVERSITY 


ELEMENTARY   ALGEBEA 


Time  :    One  Hour  and  a  Half 

Arrange  your  work  neatly  and  clearly,  beginning  each  question  on  a 
separate  page. 


1.    Simplify  the  following  expression  : 


a      b  -}-  c 
1         1 
a     b  -{-  c 


^       2  be 


2.  (a)  Write  the  middle  term  of  the  expansion  of  (a  —  by^ 
by  the  binomial  theorem. 

(b)  Find  the  value  of  aW,  if 

a  =  x^y~^     and     b  =  ^  ^~^y^y 
and  reduce  the  result  to  a  form  having  only  positive  exponents. 

3.  Find  correct  to  three  significant  figures  the  negative  root 
of  the  equation  r>  a  ^ 

1  -  -^  +  =0. 

^^  +  1^(^4.1)2 

4.  Prove  the  rule  for  finding  the  sum  of  n  terms  of  a  geomet- 
rical progression  of  which  the  first  term  is  a  and  the  constant 
ratio  is  7\ 

Find  the  sum  of  8  terms  of  the  progression 
5  +  3^  +  2|+.... 

5.  A  goldsmith  has  two  alloys  of  gold,  the  first  being  |  pure 
gold,  the  second  ^  pure  gold.  How  much  of  each  must  he  take 
to  produce  100  ounces  of  an  alloy  which  shall  be  |  pure  gold  ? 


COLLEGE   ENTRANCE   EXAMINATIONS  65 

HARVARD  UNIVERSITY 


ELEMENTAEY  ALGEBRA 


Time  :   One  Hour  and  a  Half 

1.  Solve  the  simultaneous  equations 

X  -{-  a      b 
and  verify  your  results. 

2.  Solve  the  equation  x^  —  1.6  x  —  0.23  =  0,  obtaining  the 
values  of  the  roots  correct  to  three  significant  figures. 

3.  Write  out  the  first  four  terms  of  (a  —  by. 
Find  the  fourth  term  of  this  expansion  when 


ct  =  -Vx-'yi,  &  =  V9  xyS 

expressing  the  result  in  terms  of  a  single  radical,  and  without 
fractional  or  negative  exponents. 

4.   Reduce  the  following   expression  to  a  polynomial  in  a 
^^^^'  6a' +  7  ab^-{- 12¥  1 


3 a2  -  5 a6  -  4 62        _3 5a  +  46 

19  b  19  a2 

5.  The  cost  of  publishing  a  book  consists  of  two  main  items  : 
first,  the  fixed  expense  of  setting  up  the  type ;  and,  second,  the 
running  expenses  of  presswork,  binding,  etc.,  which  may  be 
assumed  to  be  proportional  to  the  number  of  copies.  A  certain 
book  costs  35  cents  a  copy  if  1000  copies  are  published  at  one 
time,  but  only  19  cents  a  copy  if  5000  copies  are  published  at 
one  time.  Find  (a)  the  cost  of  setting  up  the  type  for  the 
book,  and  (b)  the  cost  of  presswork,  binding,  etc.,  per  thou- 
sand copies. 

REV.    ALG.  5 


66  COLLEGE   ENTRANCE   EXAMINATIONS 

HARVARD  UNIVERSITY 


ELEMENTARY   ALGEBEA 


Time  :   One  Hour  and  a  Half 

1.  Eind  the  highest  common  factor  and  the  lowest  common 
multiple  of  the  three  expressions 

a'-b'-,     a'  +  b'-,     a' -{- 2  a^b -{- 2  ab^ -\- b\ 
\j      2.    Solve  the  quadratic  equation 

x^-1.6x  +  0.3  =  0, 

computing  the  value  of  the  larger  root  correct  to  three  signifi- 
cant figures 

3.  In  the  expression 

x^  —  2xy-{'y^  —  W2(x  +  2/)  +  3, 
substitute  for  x  and  y  the  values 

u  -\-  V  -{-1  u  —  V  -{- 1 

X= ! ! y= ^ 

and  reduce  the  resulting  expression  to  its  simplest  form. 

4.  State  and  prove  the  formula  for  the  sum  of  the  first  n 
terms  of  a  geometric  progression  in  which  a  is  the  first  term 
and  r  the  constant  ratio. 

5.  A  state  legislature  is  to  elect  a  United  States  senator,  a 
majority  of  all  the  votes  cast  being  necessary  for  a  choice. 
There  are  three  candidates,  A,  B,  and  C,  and  100  members 
vote.  On  the  first  ballot  A  has  tlie  largest  number  of  votes, 
receiving  9  more  votes  than  his  nearest  competitor,  B;  but  he 
fails  of  the  necessary  majority.  On  the  second  ballot  C's  name 
is  withdrawn,  and  all  the  members  who  voted  for  C  now  vote 
for  B,  whereupon  B  is  elected  by  a  majority  of  2,  How  many 
votes  were  cast  for  each  candidate  on  the  first  ballot  ? 


COLLEGE   ENTRANCE   EXAMINATIONS  67 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY 


ALGEBEA  A 


Time  :  One  Hour  and  Three  Quarters 
1.   Factor  the  expressions  : 


oc^  -\-  x^  —  2  X. 

/)i3     _1_     /V.2    A    /y 


X 


-f  ic^  —  4  oj  —  4. 


2.    Simplify  the  expression  : 


?;2\  f^__ab-  62' 


\        a^J  \  o?     J  a?  -\- W     a^  -\-  b^ 

3.  Find  the  value  of  x  -{-  Vl  +  •'»^  when  x=  -f  -W \~  V 

4.  Solve  the  equations  : 

7x  +  6   ,   ^^      ,^__5a?-13      8y-a? 
^^^ +  2/- lt>  -  — y— -  — ^— , 

3(3a?H-4)=102/-15. 

5.  Solve  the  equations  : 

A  +  (7  =2, 

2A-B-{-2C  +  D  =  5, 
B  -\-D  =  l. 

6.  Two  squares  are  formed  with  a  combined  perimeter  of 
16  inches.  One  square  contains  4  square  inches  more  than  the 
other.     Find  the  area  of  each. 

7.  A  man  walked  to  a  railway  station  at  the  rate  of  4  miles 
an  hour  and  traveled  by  train  at  the  rate  of  30  miles  an  hour, 
reaching  his  destination  in  20  hours.  If  he  had  walked  3  miles 
an  hour  and  ridden  35  miles  an  hour,  he  would  have  made  the 
journey  in  18  hours.     E-equired  the  total  distance  traveled. 


68  COLLEGE   ENTRANCE   EXAMINATIONS 

MASSACHUSETTS   INSTITUTE   OF   TECHNOLOGY 


ALGEBEA   B 


Time  :  One  Hour  and  Three  Quarters 

1.   How  many  terms  must  be  taken  in  tlie  series  2,  5,  8,  11, 
•  •  so  that  the  sum  shall  be  345  ? 


2.  Prove  the  formula  x  =  —^-^^ — i^  for  solving  the 

2a 

quadratic  equation  ax^  -\-bx-]-  c  =  0, 

3.  Find  all  values  of  a  for  which  Va  is  a  root  of 
x^  -{- x  -\- 20  =  2  ay  and  check  your  results. 

4.  Solve   \  Q     r  ^nd  sketch  the  graphs. 

\^       X  —  y  —  Zy    j 

5.  The  sum  of  two  numbers  x  and  y  is  5,  and  the  sum  of 
the  two  middle  terms  in  the  expansion  of  (x-{-yy  is  equal  to 
the  sum  of  the  first  and  last  terms.     Find  the  numbers. 

6.  Solve  x*-2o(^-\-3x''-2x-{-l  =  0. 
(Hint  :  Divide  by  x^  and  substitute  x  +  -  =  z,) 

X 

7.  In  anticipation  of  a  holiday  a  merchant  makes  an  outlay 
of  $50,  which  will  be  a  total  loss  in  case  of  rain,  but  which 
will  bring  him  a  clear  profit  of  $  150  above  the  outlay  if  the 
day  is  pleasant.  To  insure  against*"  loss  he  takes  out  an  insur- 
ance policy  against  rain  for  a  certain  sum  of  money  for  which 
he  has  to  pay  a  certain  percentage.  He  then  finds  that  whether 
the  day  be  rainy  or  pleasant  he  will  make  $  80  clear.  What  is 
the  amount  of  the  policy,  and  what  rate  did  the  company 
charge  him  ? 


COLLEGE   ENTRANCE   EXAMINATIONS  69 

MASSACHUSETTS   INSTITUTE   OF   TECHNOLOGY 


ALGEBRA   A 


Time  :  Two  Hours 


1.    Simplify  (  m -\ —  )  +f^  +  "l  -hf  mn  H ) 

\         mj       \       nj       \  mnj 

—  [m-\ —  \\n-\--  ]( mn  -\ 

\        mj\        nj\  mn 

'    2.   Find  the  prime  factors  of 

(a)  (x-xy-\-{a^-iy-\-(l-xy. 

(b)  (2x  +  a-by-(x-a-\-b)\ 

(6)  Show  that    A/  Va^^V^-    Vx_ 

4.  Define  homogeneous  terms. 

For  what  value  of  n  is  a?"/~l  +  aj"+y"~^  a  homogeneous 
binomial  ? 

5.  Extract  the  square  root  of 

x(x-  -V2)(x  -  VS)(x  -  V18)  +  4 

6.  Two  vessels  contain  each  a  mixture  of  wine  and  water. 
In  the  first  vessel  the  quantity  of  wine  is  to  the  quantity  of 
water  as  1 :  3,  and  in  the  second  as  3  :  5.  What  quantity  must 
be  taken  from  each,  so  as  to  form  a  third  mixture  which  shall 
contain  5  gallons  of  wine  and  9  gallons  of  water? 

7.  Find  a  quantity  such  that  by  adding  it  to  each  of  the 
quantities  a,  &,  c,  d,  we  obtain  four  quantities  in  proportion. 

8.  What  values  must  be  given  to  a  and  &,  so  that 

■ — - — ■ , — ^^ — ,  and  4—0  a— 13  0  may  be  equal  r 


70 


COLLEGE   ENTRANCE   EXAMINATIONS 
MOUNT  HOLYOKE   COLLEGE 


ELEMENTAEY  ALGEBEA 


Time  :   Two  Hours 

1.  Factor  the  following  expressions : 

(a)  a?  —  bi. 

(b)  x^yh^  —  xh  —  y'^z  -\-  1, 

(c)  16{x^yy-{2x-yy, 

2.  (a)  Simplify 

_b' 

(a'  +  b')' 


b^-o? 


--4-- 


Va  -\-b     a  —  b  J 
(b)  Extract  the  square  root  of  x^  —  2x^  -\-  5  x^  —  4:X  -\-  4:, 

3.   Solve  the  following  equations  : 


(a) 

(&) 

(«) 

4.    Simplify : 

(a) 

(P) 

(c)  Find 


X      y 

l  +  i  =  13. 

x"^      y^ 

>_5aj  +  2  =  0. 
V27  x^\  =  2  -  3  V3^. 


7^54+^256  +  ^-^ 


432 
250* 


+  - 


(a  -  6)(&  -  c)      (c  -  a)(6  -  a) 
Vl9  -  8  V3. 


COLLEGE   ENTRANCE   EXAMINATIONS  71 

MOUNT   HOLYOKE   COLLEGE    (Continued) 

5.  Plot  the  graphs  of  the  following  system,  and  determine 
the  solution  from  the  point  of  intersection : 

x-2y  =  0, 
2x-Sy  =  A. 

6.  (a)  Derive  the  formula  for  the  solution  of 

ax^  -\-  bx  +  c  =  0. 

(b)  Determine  the  value  of  7n  for  which  the  roots  of 
2  a^  -^  4:X  -\-  m  =  0   are    (i)    equal,  (ii)    real,    (iii)    imaginary. 

(c)  Form  the  quadratic  equation  whose  roots  are  * 

2+V3  and  2-V3. 

i/  7.  A  page  is  to  have  a  margin  of  1  inch,  and  is  to  contain 
35  square  inches  of  printing.  How  large  must  the  page  be, 
if  the  length  is  to  exceed  the  width  by  2  inches  ? 

8.  (a)  In  an  arithmetical  progression  the  sum  of  the  first 
six  terms  is  261,  and  the  sum  of  the  first  nine  terms  is  297. 
Find  the  common  difference. 

(b)  Three  numbers  whose  sum  is  27  are  in  arithmetical 
progression.  If  1  is  added  to  the  first,  3  to  the  second,  and 
11  to  the  third,  the  sums  will  be  in  geometrical  progression. 
Find  the  numbers. 

(c)  Derive  the  formula  for  the  sum  of  n  terms  of  a  geo- 
metrical progression. 

9.  (a)  Expand  and  simplify  (2  a^  —  3  x^y. 

(b)  For  what  value  of  x  will  the  ratio  7  +  a? :  12  +  cc  be 
equal  to  the  ratio  5  ;  6  ? 


72  COLLEGE   ENTRANCE   EXAMINATIONS 

UNIVERSITY   OF    PENNSYLVANIA 


ELEMENTARY   ALGEBRA 


1.    Simplify : 


Time  :  Three  Hours 
a  -\-x     a  —  x\        4:  ax 


\a  —  X     a  -\-  xj     a^  —  Qi? 

2.  Find  the  H.  C.  F.  and  L.  C.  M.  of 

10  ah\x'  -  2  ax),  15  a'bix'  -ax -2  a^),  25  b\x''  -  a^. 

3.  A  grocer  buys  eggs  at  4  for  7  ^.  He  sells  \  of  them  at 
5  for  12  ^,  and  the  rest  at  6  for  11  ^,  making  27  ^  by  the  trans- 
action.    How  many  eggs  does  he  buy  ? 

4.  Solve  for  ^:  — ' ' =—3. 

t-\-  a-\-'b  t  -\-  a  —  b 

3  1 

5.  Find  the  square  root  of  a^—  f  a^  —  f  a^  +  ^^a-\-l. 

6.  (a)  For  what  values  of  m  will  the  roots  of  2x^  -{-3mx 

=  —2  be  equal? 
(6)  If  2  a  +  3  6  is  a  root  of  x'' -  6bx- ^a^ -^9  b""  =  0, 
find  the  other  root  without  solving  the  equation. 

7.  (a)  Solve  for  x :  V2  a?  —  3  a  +  V3  x  —  2a  =  3  Va. 
(b)  Solve  form:  1 L_  =_Jl_4.  !??:Jzl.     • 

8.  Solve  the  system :  a?^  +  2  ^/^  =  17 ;  xy  —  y^  =  2. 

9.  Two  boats  leave  simultaneously  opposite  shores  of  a 
river  2\  mi.  wide  and  pass  each  other  in  15  min.  The  faster 
boat  completes  the  trip  6|  min.  before  the  other  reaches  the 
opposite  shore.     Find  the  rates  of  tliQ  boats  in  miles  per  hour. 

10.  Write  the  sixth  term  of  f  — ^  -  ^Y  without   writing 
the  preceding  terms.  \2V.v^        x  J 

11.  The  sum  of  the  2d  and  20th  terms  of  an  A.  P.  is  10,  and 
their  product  is  23^.     What  is  the  sum  of  sixteen  terms  ? 


COLLEGE  ENTRANCE  EXAMINATIONS  73 

PRINCETON   UNIVERSITY 


ALGEBEA  A 


Time  :   Two  Hours 
Candidates  who  are  at  this  time  taking  both  Algebra  A  and  Algebra  B 
may  omit  from  Algebra  A  questions  4,  5,  and  6,  and  from  Algebra  B 
questions  1  («),  3,  and  4. 

1.    Simplify 

a^j^a^h  +  ah^    _(  a^ -[- ^  ah  -  1  Jy"  a^ -W         1 


a2  _  3  a6  -  4  62      \a? -^-^  ah -^¥   o? -1  ah +  12  h^J 

2.  (a)  Divide  a^-\-ah^-{-hi—2  ah'^—ah  by  a^  —  hi-^-ah  —  ahh 
(h)  Simplify  —-l-—^.{xWyy-\-l. 

3.  Factor:  (a)   (x" -3  xf -(2  x-6y. 

(h)  a2  +  ac-462_26c. 

4.  Solve        -J__-l- L^  +  ^_  =  0. 

x-\-l      x—1      x  —  o     X -- 5 

5.  Solve  for  x  and  y :  mx  -\-  ax  =  my  —  hy, 

x  —  y  =  a  +  h, 

6.  The  road  from  A  to  B  is  uphill  for  5  mi.,  level  for  4  mi., 
and  then  downhill  for  6  mi.  A  man  walks  from  B  to  A  in  4  hr. ; 
later  he  walks  halfway  from  A  to  B  and  back  again  to  A  in 
3  hr.  and  bh  min. ;  and  later  he  walks  from  A  to  B  in  3  hr.  and 
52  min.  What  are  his  rates  of  walking  uphill,  downhill,  and 
on  the  level,  if  these  do  not  vary  ? 

ALGEBRA   B 

1.    Solve:   (a)  —^-\ ^4-  ^  =0. 

^  ^  x-2       x-{-l        1-x 


(h)  V2aj  +  7+V3.T-18-V7aj  +  l  =  0. 
(c)  — - —  =  5-2x-x\ 


74        COLLEGE  ENTRANCE  EXAMINATIONS 

PRINCETON   UNIVERSITY   {Continued) 

2.    Solve  for  x  and  y,  checking  one  solution  in  each  problem  : 
(a)  2x-\-^y  =  l,  (h)  x'  =  x-^y, 

^  +  -  =  2.  y^  =  ^y-x, 

X      y 

^  3.  A  man  arranges  to  pay  a  debt  of  $  3600  in  40  monthly 
payments  which  form  an  A.  P.  After  paying  30  of  them  he 
still  owes  \  of  his  debt.     What  was  his  first  payment  ?     ' 

4.  If  4  quantities  are  in  proportion  and  the  second  is  a 
mean  proportional  between  the  third  and  fourth,  prove  that 
the  third  will  be  a  mean  prop,  between  the  first  and  second. 

5.  In  the  expansion  of  [2x-\ )    the  ratio  of  the  fourth 

term  to  the  fifth  is  2  : 1.     Find  x, 

6.  Two  men  A  and  B  can  together  do  a  piece  of  work  in 
12  days ;  B  would  need  10  days  more  than  A  to  do  the  whole 
work.    How  many  days  would  it  take  A  alone  to  do  the  work  ? 

ALGEBRA  TO   QUADEATICS 

1.  Simplify  (a^'V)*  •  (a^&^^-s)*  +  ^^. 

2.  Simplify  ^ + -^ + ? . 

^     ^    (a  -  h){a  -  c)      (6  -  c){h  -a)      {c-  a){c  -  b) 

3.  Factor  (a)  x^~10x^-{-9,     (b)  x^ +  2  xy -a^ -2  ay. 

(c)  (a  +  by  +  («  +  c)2  -  (c  +  df  -  (5  +  d)\ 

4.  Find  H.C.F.of  x' -x? -^2  x^  ^  x  ^^  and  (^  +  2)(a^-l). 

5.  Solve         ^_  +  ^:^  =  ^±i  +  ^-S- 


x-2      x  —  1      x  —  1      a?  — 6 
6.    The  sum  of  three  numbers  is  51 ;  if  the  first  number  be 
divided  by  the  second,  the  quotient  is  2  and  the  remainder  5 ;  if 
the  second  number  be  divided  by  the  third,  the  quotient  is 
3  and  the  remainder  2,     What  are  the  numbers  ? 


COLLEGE   ENTRANCE   EXAMINATIONS 
SMITH   COLLEGE 


75 


ELEMENTAEY   ALGEBRA 


1.    Factor  e^^  -  2  +  e-^,    x^^  -  S,  x'^  -  x- y^ -  y,    18  aV  - 
24  axy  —  10  z/l 


2.  Solve  V7  +  4  aj  +  3  V2  ^2_^5x  +  7-3  =  0. 

3.  The  second  term  of  a  geometrical  progression  is  3\^, 
and  the  fifth  term  is  ^q.     Find  the  first  term  and  the  ratio. 

4.  Solve  the  following  equations  and  check  your  results  by 
plotting : 

jx^  -^y^  —  xy  =  7, 
\x-{-y  =  4:. 


5.    Solve 


1       1  ^  243 

x^     y^       8  ' 

1+1  =  9. 

X       y      2 


6.  In  an  arithmetical  progression  d  =  —  11,  7i  =  13,  s  =  0. 
Find  a  and  1. 

7.  Expand  by  the  binomial  theorem  and  simplify : 

^2  X  y^      ^^ 


y6 


ic^V  — 6 


8.  The  diagonal  of  a  rectangle  is  13  ft.  long.  If  each  side 
were  longer  by  2  ft.,  the  area  would  be  increased  by  38  sq.  ft. 
Find  the  lengths  of  the  sides. 


76  COLLEGE   ENTRANCE   EXAMINATIONS 

SMITH   COLLEGE 


ELEMENTARY   ALGEBRA 


1.  Find  the  H.  C.  F.  of  8  aj^  -  27,  32  x'  -  243,  and  Ga^-dx" 
+  4  a;  -  6. 

2.  Solve: 

(a)  (2  a;  4- 5)-^  +  31(2  a^  +  5)"4  =  32. 

(b)  (x  -  l)i  +  (3  oj  +  1)*  =  4. 

3.  A  farmer  sold  a  horse  at  $  75  for  which  he  had  paid  x 
dollars.     He  realized  x  per  cent  profit  by  his  sale.     Find  x. 

4.  Find  the  13th  term  and  the  sum  of  13  terms  of  the 
arithmetical  progression 

V2-1         V2  1 

2       '  2  '       2(V2-iy 

5.  The  difference  between  two  numbers  is  48.  Their  arith- 
metical mean  exceeds  their  geometrical  mean  by  18.  Find  the 
numbers. 

6.  Expand  by  the  binomial  theorem  and  simplify 
1     1^3 

X    y    2' 
1  +  1  =  5. 

x^      y'      4  ' 

8.  Solve  the  following  equations  and  check  the  results  by 
finding  the  intersections  of  the  graphs  of  the  two  equations : 

(x'^  =  4:y, 
\x  +  2y  =  4.. 


7.    Solve: 


COLLEGE   ENTRANCE   EXAMINATIONS  77 

VASSAR   COLLEGE 


ELEMENTARY  AND   INTEEMEDIATE   ALGEBRA 


Answer  any  six  questions. 

1.  Find  the  product  of 

\   ,  2a     5a'^\        -i  /o     Sa  ,  a'^\ 
l+_-_jand(^2-_  +  -J. 

2.  Resolve  into  linear  factors : 

(a)  4x2-25;  (b)  6x\-x-12',  (c)  a^b'' -{- 1  -  a"  -  b^ ; 
(d)  f+(x-3)f-{3x-2)y-{-2x. 

3.  Reduce  to  simplest  form  : 

(«)   1^  +  ^ ^-  (^)  |-(a^)4jix(4r')i. 

± __±    ^ y    i__ 

X      y  X  y 

4.  (a)  Divide  x^  —  x~^  by  x^  —  x~^. 

(b)  Find  correct  to  one  place  of  decimals  the  value  of 
V5  +  V7. 
2-V3  * 

5.  (a)  If  -  =  ^,  show  that  ^^-±^  =  ^. 
^  ^       b     d'  b^  +  d^     bd 

(b)  Two  numbers  are  in  the  ratio  3  :  4,  and  if  7  be  subtracted 

from  each  the  remainders  are  in   the  ratio   2  : 3.     Find  the 

numbers. 

6.  Solve  the  equations  : 

^  ^    ~2~     ^-3      ~6~  ^'^    [x-^y  =  20. 

(b)   11  a;2  -  11^  =  9  oj. 

7.  A  field  could  be  made  into  a  square  by  diminishing  the 
length  by  10  feet  and  increasing  the  breadth  by  5  feet,  but  its 
area  would  then  be  diminished  by  210  square  feet.  Find  the 
length  and  the  breadth  of  the  field. 


78  COLLEGE  ENTRANCE   EXAMINATIONS 

VASSAR   COLLEGE 


ELEMENTAEY   AND   INTEEMEDIATE   ALGEBRA 


Answer  six  questions,  including  No.  5  and  No.  7  or  8.     Candidates  in 
Intermediate  Algebra  will  answer  Nos.  5-9. 

1.  Find  two  numbers  whose  ratio  is  3  and  such  that  two 
sevenths  of  the  larger  is  15  more  than  one  half  the  smaller. 

2.  Determine  the  factors  of  the  lowest  common  multiple  of 
3  x'  {x^  -  y^),  15  (x'  -  2  xY  +  y'),  and  10  y  (a;*  +  ^Y  +  y^)- 

3.  Find  to  two  decimal  places  the  value  of 

-^4  oT^  +  h^  Va6^,  when  a  =  -  32  and  6  =  -  8. 

J     4.    Solve  the  equations  :     2x-\-5  y  =  85, 

2y-i-5z=103, 

2z-i-5x  =  57. 

5.    Solve  any  3  of  these  equations : 


(a)  x''  +  U-15x  =  0.       (c)  x'-^Sx--V4.x'-h32x-\-12  =  21. 

..   2_^^_^_223  ...       5  8     ^       12 

^^x     5      20       30*         ^  ^  x-{-l      x-2     4.0  -  2  x' 
X/'  6.    The  sum  of  two  numbers  is  13,  and  the  sum  of  their 

cubes  is  910.     Find  the  smaller  number,  correct  to  the  second 

decimal  place. 

7.    The  sum  of  9  terms  of  an  arithmetical  progression  is  46 ; 

the  snm  of  the  first  5  terms  is  25.    Find  the  common  difference. 
'     8.   Explain  the  terms,  and  prove  that  if  four  numbers  are  in 

proportion,  they  are  in  proportion  by  alternation,  by  inversion, 

and  by  composition.     Find  x  when    ' 

3  —  X      M)  —  x^ 

9.    Find  the  value  of  x  in  each  of  these  equations : 

(a)  1x^-3x^  =  2,  (h)  (x^J^2f-\-        ^        =4.x'-\-S, 

■\/x^  -f-  2 


COLLEGE   ENTRANCE   EXAMINATIONS  79 

YALE   UNIVERSITY 


ALGEBEA  A 


Time  :  One  Hour 

Omit  one  question  in  Group  11  and  one  in  Group  III.     Credit  will  be 
given  for  six  questions  only. 

Group  I 

1.  Resolve  into  prime  factors  :  (a)  6  x^  —  7  x  —  20; 

(b)     (x''-5xy-2(x'-5x)-24;  (c)  a' -{- 4.  a^  ^  16. 

2.  Simplify  /^5_a^-19^^\     /3_a-5a.Y 

3.  Solve  2(0^-7)      ^2-x_^±3^Q^ 

a;2  4_  3  ^^  _  28      4  -  a;      x-{-7 

Oroiip  II 

4.  Simplify  — _  "*"  "^  ,  and  compute  the  value  of  the  frac- 

V2-V12 

tion  to  two  decimal  places. 


5.    Solve  the  simultaneous  equations 


-i  +  22/-^  =  |, 


2x  2  —  2/  2  = 


3* 


Group  III 

6.  Two  numbers  are  in  the  ratio  of  c  :  d.  If  a  be  added  to 
the  first  and  subtracted  from  the  second^  the  results  will  be  in 
the  ratio  of  3  :  2.     Find  the  numbers. 

7.  A  dealer  has  two  kinds  of  coffee,  worth  30  and  40  cents 
per  pound.  How  many  pounds  of  each  must  be  taken  to  make 
a  mixture  of  70  pounds,  worth  36  cents  per  pound  ? 

8.  A,  B,  and  C  can  do  a  piece  of  work  in  30  hours.  A  can 
do  half  as  much  again  as  B,  and  B  two  thirds  as  much  again  as 
C.     How  long  would  each  require  to  do  the  work  alone  ? 


80  COLLEGE   ENTRANCE   EXAMINATIONS 

YALE   UNIVERSITY 


ALGEBEA  B 


Time  :    One  Hour 

Omit  one  question  in  Group  I  and  one  in  Group  II.  Credit  will  be 
given  for  Jive  questions  only. 

Group  I 

,/-.     ai       x-\-  a  ,  x-\-h      5 
i/  1.    Solve     ^     +  =7z' 

X  -\-  b      x-i-  a      2 

( xY-\-2Sxy-AS0  =  0, 

2.  Solve  the  simultaneous  equations  ■{  ^ 

l2x+y=ll. 

Arrange  the  roots  in  corresponding  pairs. 

3.  Solve  3x~^+20x~^=z32. 

Group  II 

4.  In  going  7500  yd.  a  front  wheel  of  a  wagon  makes  1000 
more  revolutions  than  a  rear  one.  If  the  wheels  were  each  1  yd. 
greater  in  circumference,  a  front  wheel  would  make  625  more 
revolutions  than  a  rear  one.     Find  the  circumference  of  each. 

5.  Two  cars  of  equal  speed  leave  A  and  B,  20  mi.  apart,  at 
different  times.  Just  as  the  cars  pass  each  other  an  accident 
reduces  the  power  and  their  speed  is  decreased  10  mi.  per  hour. 
One  car  makes  the  journey  from  A  to  B  in  56  min.,  and  the 
other  from  B  to  A  in  72  min.     What  is  their  common  speed  ? 

Group  IIL 
/  6.    Write  in  the  simplest  form  the  last  three  terms  of  the 
expansion  of  (4  a^  —  a^x^y. 
.  7.    (a)  Derive  the  formula  for  the  sum  of  an  A.  P. 

(b)  Find   the    sum  to  infinity  of   the    series    1,  —  ^,  ^, 
—  i,  •...     Also  find  the  sum  of  the  positive  terms. 


,  14  DAY  USE 

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